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Complex Continued Fractions

Theoretical Aspects of Hurwitz’s Algorithm

PhD Thesis

Gerardo Gonzalez Robert

Supervisor

Simon Kristensen

Department of Mathematics

Aarhus University

Denmark

September 2018

Complex Continued Fractions

by


Denmark2018

Ya, ya me esta gustando mas de lo normal,Todos mis sentidos van pidiendo mas,

Esto hay que tomarlo sin ningun apuro.

–Ramon Luis Ayala Rodrıguez, Despacito, 2017

Abstract

Inspired by their remarkable properties, mathematicians have sought in sev-eral contexts objects similar to regular continued fractions. Hurwitz Con-tinued fractions (HCF), proposed by Adolf Hurwitz in 1887, give a naturalanalogue in the complex plane. The HCF of a complex number ζ is a se-quence of Gaussian integers obtained from a specific algorithm. Among thesimilarities between HCF and regular continued fractions we can find theirrationality of a complex number as a necessary and sufficient condition forits HCF to be infinite. In this case, there even are combinatorial conditionson the HCF implying transcendence. The usual characterization of quadraticsurds in terms of periodic continued fractions also holds. However, E. Galois’theorem on purely periodic continued fractions is not true unless additionalrestrictions are imposed.

HCF can be understood through a dynamical perspective. With a suit-ably defined Gauss map, HCF give rise to an ergodic dynamical system.Although the corresponding shift space has a complicated structure, someclassical Hausdorff dimension results still hold. For example, the set of com-plex numbers with a bounded HCF has full Hausdorff dimension (proved byV. Jarnık in 1929 for R) and the Hausdoff dimension of the set of complexnumbers with a HCF tending to ∞ is half of that of the ambient space’s(proved by I.J. Good in 1941 for R). The theory of HCF is far from beingcompleted, and its similarities and discrepancies with their real counterpartstem new problems.

The present thesis is a survey of new and old results on HCF. In thepreface, a detailed list of all the novel aspects is given. Additionally, threeoriginal research papers are included as appendices. The first two deal withHCF theory. The third one — applying the geometry of regular continuedfractions— contains a simple proof of P. Bengoechea’s quantitative improve-ment of Serret’s Theorem on equivalent numbers.

Resume

Inspireret af deres bemærkelsesværdige egenskaber har matematikere iflere sammenhænge søgt objekter svarende til regulære kædebrøker. Hurwitz-kædebrøker (HCF), der blev introduceret af Adolf Hurwitz i 1887, giver ennaturlig analog i den komplekse plan. HCF af et komplekst tal ζ er en følgeaf gaussiske heltal udregnet ved en specifik algoritme. Blandt lighedernemellem HCF og regulære kædebrøker kan vi finde, at irrationalitet af etkomplekst tal er en nødvendig og tilstrækkelig begtingelse for, at HCF eruendelig. I dette tilfælde er der endda kombinatoriske betingelser pa talletsHCF der garanterer tallets transcendens. Den sædvanlige karakterisering afkvadratiske irrationale som periodiske kædebrøker holder ligeledes. Pa denanden side er E. Galois sætning om fuldstændigt periodiske kædebrøker ikkesand, med mindre vi indfører yderligere restriktioner.

HCF kan forstas fra et dynamisk synspunkt. Med en passsende defineretGauss-afbildning, giver HCF anledning til et ergodisk dynamisk system.Selvom det tilhørende skift-rum har en kompliceret struktur, holder visseklassiske resultater om Hausdorff-dimension fortsat. For eksempel har mæng-den af komplekse tal med en begrænset HCF maksimal Hausdorff-dimension(vist af V. Jarnk i 1929 for R), og Hausdorff-dimensionen af mængden af kom-plekse tal for hvilke HCF gar mod ∞ er halvdelen af det omgivende rumsdimension (vist af I. J. Good i 1941 for R). Teorien om HCF er langt frafuldstændig, og fra dennes ligheder og uoverensstemmelser med dens reellemodpart vokser nye problemer.

Den foreliggende afhandling er en undersøgelse af nye og gamle resultaterom HCF. I forordet findes en detajeret liste over alle nye aspekter. Desudener tre originale forskningsartikler inkluderet som appendices. De to førsteomhandler HCF-teori. Den tredje, der anvender teorien om geometriske reg-ulære kædebrøker, indeholder et simpelt bevis for P. Bengochea’s kvantitativeforbedring af Serrets sætning om ækvivalente tal.

Preface

Regular continued fractions have a long standing tradition of noble service tonumber theory; in particular, to diophantine approximation. Among otherproperties, they provide an algorithmic solution to the problem of finding thebest rational approximation of a given real number. Unlike other representa-tions, like b-ary or β-expansions, regular continued fractions do not dependon any number other than the represented one. Regular continued fractionsalso express right away intrinsic properties of the numbers they represent,like being badly approximable or a quadratic irrational, for example. In viewof these nice properties, it is natural to ask for the existence of a similarprocess in other contexts. Several answers have been given depending on thefeatures that are intended to extend. This text focuses on a particular con-tinued fraction algorithm for complex numbers suggested by Adolf Hurwitzin 1887, the Hurwitz Continued Fractions (HCF). This document is a resultof my work in the PhD Programme in Mathematics at Aarhus Universityunder the supervision of Dr. Simon Kristensen.

Although Hurwitz Continued Fractions are an old subject, they have notgot much attention by themselves until 2006, when D. Hensley constructed al-gebraic numbers of degree 4 with bounded partial quotients. Unfortunately,D. Hensley’s work sheds no new light on the corresponding conjecture inthe real numbers. A considerable amount of research has been done on theergodic theory of some dynamical systems that include Hurwitz ContinuedFractions. One of our main goals is to survey new and old results, generaliza-tions, and common techniques and strategies under a unified approach. Sincewe contrast constantly the theory of regular and Hurwitz continued fractions,we assume the reader is acquainted with regular continued fractions.

The thesis is divided into six chapters and five appendices. Each chapterstarts with an abstract, a comment on the notation and a citation of themain references. In Chapter I, the framework of complex continued frac-tions introduced by S.G. Dani and A. Nogueira is presented as well as somebasic properties (like representation of complex numbers and convergence).Within this framework, Hurwitz Continued Fractions are properly defined

v

Preface

and their approximation properties are studied. In Chapter 2, we discuss thealgebraic properties of HCF through analogues of Euler-Lagrange Theorem,Galois Theorem on purely periodic continued fractions, and Y. Bugeaud’sconstruction of transcendental numbers through automatic sequences. Wealso reproduce W. Bosma and D. Gruenewald extension of D. Hensley’s re-sult. In Chapter 3, we discuss the ergodic theory of HCF. In particular, weassociate a natural dynamical system and construct a measure µ equivalentto the Lebesgue measure making the system ergodic. We discuss briefly twonotions that generalize HCF, one of them is Iwasawa Continued Fractionsand the other one is Fibred Systems. In Chapter 4, we explore the Hausdorffdimension of some sets defined by imposing restrictions on the HCF expan-sion. We stress along the chapter the importance of the bounded distortion ofthe functions involved in the dynamical system associated to HCF. In Chap-ter 5, we give a short overview of the complex continued fraction algorithmproposed by J. Hurwitz, also within the framework of Dani and Nogueira.Chapter 6 contains a short list of related problems and a brief description ofongoing research.

In the first two appendices, we just set the notation and record some basicresults. The last three appendices are the research papers I finished duringthe PhD programme:

1. “Purely Periodic and Transcendental Complex Continued Fractions” (sub-mitted),

2. “Good’s Theorem for Hurwitz Continued Fractions” (submitted),

3. “Geometric Continued Fractions and Equivalent Numbers”.

The third paper is not related with the rest of the thesis. It is the resultof a seminar on the geometry of continued fractions held with Dr. SimonKristensen and Dr. Morten Hein Tiljeset.

Besides suggesting a unified notation for the treatment of HCF, the fol-lowing are the innovations of the thesis:

1. A small gap in the proof of Theorem 1.2.4 is filled for HCF. A minormistake was also corrected in the proof of Theorem 2.2.3.

2. We introduce the notion of irregular numbers and sequences. Althoughthe idea is clearly present in previous works, like the ones of D. Hensley,S. Tanaka, H. Nakada, and more, they were completely dismissed forbeing a null set.

3. Although simple and natural, we have not found Theorem 2.1.3 in theliterature.

vi

Preface

4. Theorems 2.3.2 and 2.4.1 are new (see [Gon18-01]).

5. The argument of the proof of the First Lakein Theorem (Theorem 1.2.6)is the original one, but the exposition is drastically modified. Somedetails are added and, hopefully, a much clearer proof is attained.

6. We have not found the first part of Proposition 3.2.6 in the availableliterature.

7. The Generalized Jarnık Lemmas, Good’s Theorem for Hurwitz Con-tinued Fractions, and Theorem 4.5.2 are new.

8. The convergence of the Hurwitz-Tanaka continued fraction of everycomplex irrational is obtained as an immediate consequence of a resultby S.G. Dani and A. Nogueira. The proof of Theorem 5.1.1 is signifi-cantly shorter and simpler than the previously known (see [Os16]).

All the figures were done with TiKz.

Acknowledgements. I thank Dr. Simon Kristensen for his supervisionand enormous help with mathematical and non-mathematical issues evenbefore my arrival to Denmark. I thank Dr. Yann Bugeaud for receiving meduring a very fruitful semester at the Institut de Recherche MathematiqueAvancee in Strasbourg, France. I want to thank Dr. Tomas Persson forhis comments and, as Dr. Oleg German and Dr. Oleg Karpenkov, for hisdisposition. I want to thank everyone at Aarhus University’s MathematicsDepartment for providing an unbeatable working environment with a warmhuman touch.

My research would not have been possible without the funding of theConsejo Nacional de Ciencia y Tecnologıa (grant: 410695), and the additionalgrants given by the Secretarıa de Educacion Publica and Aarhus University(Projekt: 25763, Aktivitet: 26248).

I thank Christina for her great support, particularly while writing thisthesis. I thank the friends and family I got to know at Aarhus Universityand at Resedavej for making these years unforgettable. I thank Flavie andYann Kaision and their family for the nice time in Strasbourg. Finally, Ithank my friends and family from Mexico for their backing.

vii

Preface

viii

Contents

Preface v

1 Basic Properties 11.1 Complex Continued Fractions . . . . . . . . . . . . . . . . . . . 21.2 Hurwitz Continued Fractions . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Laws of Succession . . . . . . . . . . . . . . . . . . . . . 61.2.2 Growth of Denominators . . . . . . . . . . . . . . . . . . 121.2.3 Quality of Approximation . . . . . . . . . . . . . . . . . 191.2.4 Serret’s Theorem . . . . . . . . . . . . . . . . . . . . . . 27

1.3 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Algebraic properties 292.1 Complex Algebraic Numbers of Degree 2 . . . . . . . . . . . . 29

2.1.1 Purely Periodic HCF . . . . . . . . . . . . . . . . . . . . 312.2 Complex Algebraic Numbers of Degree Greater than 2 . . . . 33

2.2.1 Generalized Circles . . . . . . . . . . . . . . . . . . . . . 332.2.2 Circles of a complex irrational . . . . . . . . . . . . . . 34

2.3 Transcendental numbers . . . . . . . . . . . . . . . . . . . . . . 372.3.1 Combinatorics on words . . . . . . . . . . . . . . . . . . 372.3.2 Complex Transcendental Numbers . . . . . . . . . . . . 39

2.4 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Ergodic Theory of Hurwitz Continued Fractions 433.1 Geometric observations . . . . . . . . . . . . . . . . . . . . . . . 443.2 An ergodic measure . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.1 Consequences and related results . . . . . . . . . . . . . 493.3 General frameworks . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.1 Fibred Systems . . . . . . . . . . . . . . . . . . . . . . . 563.3.2 Iwasawa Continued Fractions . . . . . . . . . . . . . . . 58


ix

CONTENTS CONTENTS

4 Hausdorff Dimension Theory 614.1 Hausdorff Dimension. Definitions and Elementary Properties 62

4.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Jarnık’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.1 Jarnık’s Strategy . . . . . . . . . . . . . . . . . . . . . . 654.2.2 Extensions, abstractions, and generalizations . . . . . . 67

4.3 A word on Schmidt Games . . . . . . . . . . . . . . . . . . . . . 714.4 Hausdorff Dimension for HCF Sets . . . . . . . . . . . . . . . . 73

4.4.1 Measure and Dimension of BadC . . . . . . . . . . . . . 734.4.2 Good’s Theorem for HCF . . . . . . . . . . . . . . . . . 75

4.5 Independent Blocks . . . . . . . . . . . . . . . . . . . . . . . . . 764.5.1 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . 774.5.2 Proof of Theorem 4.4.3 . . . . . . . . . . . . . . . . . . . 80

4.6 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Hurwitz - Tanaka Continued Fractions 835.1 Definition and convergence . . . . . . . . . . . . . . . . . . . . . 835.2 Laws of Succession . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3 The Ergodic Theory . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3.1 The Dual Algorithm . . . . . . . . . . . . . . . . . . . . 885.3.2 The Natural Extension . . . . . . . . . . . . . . . . . . . 905.3.3 The Ergodic Measure . . . . . . . . . . . . . . . . . . . . 91


6 Current and Future Research 956.1 Current Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.1.1 A Kaufman Measure in the Complex Plane . . . . . . . 956.1.2 A Generalization of Good’s Theorem . . . . . . . . . . 96

6.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Appendix A 99

Appendix BErgodic Theory 1016.3 Rokhlin’s Natural Extension . . . . . . . . . . . . . . . . . . . . 104

Bibliography 107

Papers 113

x

Chapter 1

Basic Properties

In this chapter, we introduce the complex continued fraction framework de-veloped by S. G. Dani and A. Nogueira in their 2014 paper [DaNo14]. Withintheir framework, we discuss some general properties on representation ofcomplex numbers. Afterwards, we consider the complex continued fractionalgorithm proposed by Adolf Hurwitz as a particular case. We explore someapproximation properties of Hurwitz’s algorithm and conclude with a char-acterization of badly approximable complex numbers (defined below).

Main references. For the theory of complex continued fractions: [Da15],[DaNo14], [La73], [La74a], [La74b], [H87], [He06]. For the classical theory ofregular continued fractions: [Kh].

Notation. For any A ⊆ C and z ∈ C we write

z +A ∶= z + a ∶ a ∈ A, zA ∶= za ∶ a ∈ A.For A ⊆ C, A is the interior of A, ClA is the closure of A, A∗ ∶= A∖0, andA′ ∶= A ∖Q(i).

Denote the unit disc by D = z ∈ C ∶ ∣z∣ < 1 and D = ClD. For every z ∈ Cand r > 0 we write

D(z, r) ∶= z + rD, D(z, r) ∶= z + rD, D(z) ∶= z +D, D(z) ∶= z +D,E(z, r) ∶= C ∖D(z, r), E(z, r) ∶= C ∖D(z, r), E(z) ∶= C ∖D(x),

E(z) ∶= C ∖D(z).The Golden Ratio is denoted by φ = 1+√5

2 . For m,n ∈ Z, [m..n] = j ∈ Z ∶m ≤ j ≤ n.

1

1.1 Complex Continued Fractions Basic Properties

1.1 Complex Continued Fractions

Let z be a complex number. A sequence (wn)n≥0 in C (possibly finite) is aniteration sequence of z if

w0 = z, ∀n ∈ N ∣wn∣ ≥ 1, wn − 1

wn+1

∈ Z[i] ∖ 0, (1.1)

We call (wn)n≥0 degenerate if ∣wn∣ = 1 for some n ∈ N and non-degenerateotherwise. The partial quotients or elements is the sequence (an)n≥0 inZ[i] ∖ 0 given by

∀n ∈ N0 an = wn − 1

wn+1

. (1.2)

Following [DaNo14], we call the sequences (pn)n≥0, (qn)n≥0 given by

(p−2 p−1

q−2 q−1) = (0 1

1 0) , ∀n ∈ N0 (pn+1

qn+1) = (pn pn−1

qn qn−1)(an+1

1) . (1.3)

the Q-pair of z. For a0, a1, . . . , an ∈ Z[i] we will write

⟨a0;a1, a2, . . . , an⟩ = a1 + 1

a2 + 1

⋱ + 1

an

.

We reserve the notation [a0;a1, a2, . . .] for Hurwitz Continued Fractions (de-fined below). Note that if z ∈ C has a finite iteration sequence (wn)Nn=0 andelements (an)Nn=0, then z = ⟨a0;a1, . . . , an⟩.

Several classical identities of continued fractions depend on the recur-rence relations defining (pn)n≥0 and (qn)n≥0. For convenience, we recall themwithout a proof.

Proposition 1.1.1. Let z be a complex number, (wn)n≥0 an iteration se-quence of z, (an)n≥0 its sequence of elements, ((pn)n≥0, (qn)n≥0) its Q-pair,and N ⊆ N0 the set of indices where the iteration sequence is defined. Then,for every n ∈ N we have

pnqn

= ⟨a0;a1, . . . , an⟩ (1.4)

qnpn−1 − qn−1pn = (−1)n (1.5)

qnz − pn = (−1)nw1w2⋯wn , (1.6)

z(qnwn+1 − qn−1) = pnwn+1 − pn−1. (1.7)qnqn−1

= ⟨an;an−1, . . . , a1⟩ if qn ≠ 0. (1.8)

2

Basic Properties 1.1 Complex Continued Fractions

We refer to (1.8) as the mirror formula.

Remark. Whenever ∣wn∣ > 1, (1.6) forces qm ≠ 0 for m ≥ n.

Suppose z ∈ C is a complex irrational, that is z ∈ C ∖Q(i). Then, anyiteration sequence of z has to be infinite. However, we cannot say a priori ifthe equality

z = ⟨a0;a1, a2, a3, . . .⟩ (1.9)

holds (the right side understood as the limit when n→∞ of ⟨a0;a1, a2, . . . , an⟩).Under mild conditions, the infinite continued fraction arising from an itera-tion sequence actually represents z; i.e., (1.9) is true.

Theorem 1.1.2 (Dani and Nogueira (2011)). Let z be a complex irrational.If (zn)n≥0 is a non-degenerate iteration sequence of z, then

limn→∞

pnqn

= z.Proof. In view of (1.6), it suffices to show that lim supn ∣wn∣ > 1. We shall as-sume for contradiction that the inequality fails, this is equivalent to limn ∣wn∣ =1. Define the set

R = z ∈ C ∶ ∣z∣ = 1, ∃w ∈ Z[i] d(z,w) = 1 = z ∈ C ∶ z12 = 1,then, the recurrence satisfied by (wn)n≥0 yields

∣wn − an∣ = 1∣wn+1∣ → 1 as n→∞,so d(wn,R)→ 0 when n→∞. Let (ρn)n≥0 be a sequence in R satisfying

limn→∞ ∣wn − ρn∣ = 0; (1.10)

hence,limn→∞ ∣ρn − an∣ = 1,

Consider the sequences (ζn)n≥1, (βn)n≥1 given by

∀n ∈ N ζn ∶= wn − ρn, βn ∶= ρn − an.Note that βn ∈ R for sufficiently large n, because R is finite, βn → 1 as n→∞,and an ∈ Z[i]. Then, the identities w−1

n+1 = wn − an = ζn + βn, (1.10), and thedefinitions of ζn and βn give

wn+1 = 1

βn+ −ζnβn(βn + ζn) = ρn+1 + εn+1 = βn+1 + ζn+1, (1.11)

3

1.1 Complex Continued Fractions Basic Properties

where εn → 0 as n→∞. For large n we have βn ∈ R and

∣ ζnβn(βn + ζn)∣ ≤ ∣ζn∣

1 − ∣ζn∣ ≤ ∣ζn∣→ 0 when n→∞. (1.12)

Thus, the finiteness of R forces the next equalities to eventually hold

βn+1 = ρn+1 = 1

βn, ζn+1 = −ζn

βn(βn + ζn) .By (1.12), for large n we have ∣ζn+1∣ ≥ ∣ζn∣ and, since ζn → 0, the sequence(ζn)n≥0 ends up being constant and equal to 0. Therefore, in view of (1.11),wn = ρn and ∣wn∣ = 1 for large n, contradicting the non-degeneracy hypothesis.Hence, lim supn ∣wn∣ > 1 and z = ⟨a0;a1, a2 . . .⟩.

A simple way to construct iteration sequences is via choice functions. Wecall f ∶ C∗ → Z[i] a choice function if

∀z ∈ C∗ f(z) ∈ D(z).The fundamental set of f , Ff , is

Ff ∶= Clz − f(z) ∶ z ∈ C∗.For any z ∈ C∗ we define the iteration sequence (wn)n≥0 = (wn(f, z))n≥0 by

w0 = z, ∀n ∈ N wn = 1

wn−1 − f(wn−1) , (1.13)

as long as wn−1 ≠ f(wn−1), in which case we only keep (w0, . . . ,wn−1) andsay that the algorithm terminates. The f-sequence of z is the sequenceof elements of z associated to (wn)n≥0, they satisfy an = f(wn).Theorem 1.1.3. Let f be a choice function such that Ff ⊆ D(0, r) for some0 < r < 1. Let ζ be a complex number and (an)n≥0 its f -sequence.

i. If ζ ∈ Q(i), then (an)n≥0 is finite, say (an)n≥0 = (a0, a1, . . . , aN), and

ζ = ⟨a0;a1, . . . , aN⟩ .ii. If ζ ∈ C ∖Q(i), then (an)n≥0 is infinite and

ζ = limn→∞ ⟨a0;a1, a2, . . . , an⟩ .

4

Basic Properties 1.2 Hurwitz Continued Fractions

Proof. i. Write ζ = ab ∈ Q(i) with a, b ∈ Z[i] coprime. For any n ∈ N such

that an is defined and pn/qn ≠ ζ, we have that

1∣b∣ ≤ ∣qnab− pn∣ < rn+1,

so n must be bounded and (an)n≥0 is finite. Since the algorithm termi-nates, for some N ∈ N we have aN = f(zN) and ζ = ⟨a0;a1, . . . , aN⟩ by(1.13).

ii. It follows directly from Theorem 1.1.3.

Let f be a choice function. The family of sets f−1[a] ∶ a ∈ Z[i] is acountable partition of C∗ satisfying

⋃a∈Z[i]−a +Ga ⊆ D. (1.14)

On the other hand, we can construct a choice function starting from a count-able partition G = Ga ∶ a ∈ Z[i] such that (1.14) holds. We just considerf ∶ C∗ → Z[i] to be given by the condition f(z) ∈ Gf(z). Suppose that Gadditionally satisfies

∀z ∈ C∗ z − f(z) ∈ G0 ∪ 0. (1.15)

We define the f-Gauss map Tf ∶ G∗0 → G0 by

∀z ∈ G0 Tf(z) = 1

z− f (1

z) .

In this case, to each z ∈ C′ we represent its orbit under Tf by (zn)n≥0. Thatis, z0 = z and zn = Tf(zn−1) for each n ∈ N.

Although restrictive, the condition (1.15) is not artificial. It holds, forexample, if we consider a sub-lattice Λ ⊆ Z[i] with [Z[i] ∶ Λ] ≤ 2 and takeas G the translations of any fundamental domain of C/Λ contained in D. Infact, the two algorithms that will concern us satisfy (1.15).

1.2 Hurwitz Continued Fractions

Adolf Hurwitz proposed in 1887 a complex continued fraction algorithmwhich shares some non-trivial and highly desirable properties with the realregular continued fractions. In a few words, his algorithm is a straight for-ward generalization of the nearest integer continued fraction to the complex

5

1.2 Hurwitz Continued Fractions Basic Properties

plane. With a suitable choice function, A. Hurwitz’s algorithm can be un-derstood as an example of the iteration sequence theory.

The nearest Gaussian integer function [⋅] ∶ C→ Z[i] associates to eachcomplex number z the Gaussian integer closest to z. In case of a tie, it takesthe one with greatest real or imaginary part. Thus, the inverse image of 0under [⋅] is

F ∶= z ∈ C ∶ −1

2≤Rz,Iz < 1

2 , (1.16)

For any z ∈ F∗ define— as long as the operations make sense— the sequences(an)n≥0, (zn)n≥1 by

z1 ∶= z, ∀n ∈ N zn+1 = T (zn),a0 ∶= 0, ∀n ∈ N an = [z−1

n ] ,where T is the corresponding f -Gauss map. The sequence (an)n≥0 is theHurwitz continued fraction of z.

Definition 1.2.1. The Hurwitz Continued Fraction (HCF) of z ∈ Cis its [⋅]-sequence. If (an)n≥0 is the HCF of z, we write z = [a0;a1, a2, . . .]when the sequence is infinite and z = [a0;a1, a2, . . . , an] when it is finite andcomprises n + 1 elements.

From now on, except when explicitly stated, all continued fractions con-sidered are HCF. Let z be a complex irrational and ((pn)n≥0, (qn)n≥0) itsQ-pair. The hypothesis of Theorem 1.1.3 are clearly met (r = 2− 1

2 ), soz = [a0;a1, a2, . . .] is indeed true. Moreover, by ∣z−1

n ∣ ≥ √2 and (1.6), we

even know the convergence is exponentially fast:

∀n ∈ N ∣qnz − pn∣ ≤ 2−n2 .It is not clear, though, whether ∣qn∣ tends to ∞ or not, or even if it is strictlyincreasing. In order to show these important features, we need to study theassociated space of sequences of Gaussian integers associated to the HCFprocess.

1.2.1 Laws of Succession

Regular continued fractions give a topological conjugacy between the dynam-ical systems ([0,1]∖Q, TR) and (NN, σ). Here and in the following, TR is theclassical Gauss map x↦ x−1−[x−1] where [x−1] is the integral part of x−1 andσ is the shift map: σ((xn)n≥1) = (xn)n≥2. In the HCF context, the structureof the the sequence space is much more complicated.

6


A cylinder of depth n ∈ N is a set of the form

Cn(a) = z = [0; b1, b2, . . .] ∶ a1 = b1, . . . , bn = an,where a is a sequence in Z[i]. In Figure 1.1 we show the partition of Finduced by cylinders of depth 1. Note that for a ∈ Z[i] we have C1(a) ≠ ∅ ifand only if a belongs to

I ∶= a ∈ Z[i] ∶ ∣a∣ ≥ √2.

1

−0.5 0.5

−0.5

0.5

0

1 − i−1 − i

1 + i−1 + i

−2 − i

−1 − 2i −2i 1 − 2i

2 − i

2

2 + i

1 + 2i

2 + 2i3i

2i

−1 + 2i

−2 + i

−2 −3−3 − i

−3 + i

−2 + 2i−1 + 3i 1 + 3i

3 + i

3

3 − i3 − 2i

2 − 2i−3i

−2 − 2i −1 − 3i

Figure 1: Partition of F.

Figure 1.1: Partition C1(a) ∶ a ∈ I of F.

Terminology

A sequence (an)n≥0 in Z[i] is valid or admissible if it is the Hurwitz Contin-ued Fraction of some z ∈ F. The space of admissible sequences is denoted byΩHCF. Clearly, a sequence a in Z[i] belongs to ΩHCF if and only if Cn(a) ≠ ∅for all n. We will call the rules that determine the membership of a sequencea ∈ Z[i]N to ΩHCF the laws of succession. We admit that the definition israther loose, but no confusion shall arise from it.

A non-empty set is a maximal feasible set if it has the form T n[Cn(a)]for some a ∈ ΩHCF and some n ∈ N. A maximal feasible set is regular if it

7


has non-empty interior, otherwise it is irregular and strongly irregularif it has only one point. A sequence a ∈ ΩHCF is regular (resp. irregular,strongly irregular) if T n[Cn(a)] is regular (resp. irregular, strongly irreg-ular). A complex irrational z ∈ F is regular (resp. irregular, stronglyirregular) if its HCF is regular (resp. irregular, strongly irregular). Theset of regular sequences (resp. irregular sequences, regular irrationals in F,irregular irrationals in F) is denoted by ΩHCF

R (resp. ΩHCFI , FR, FI).

Determination of the laws

The laws of succession are determined by the behaviour of complex inversion.Observe that F−1 = z−1 ∶ z ∈ F∗ can be expressed as

F−1 = E(1) ∩E(i) ∩E(−1) ∩E(−i)(see Figure 1.2). It is clear that for a ∈ Z[i] we have (a+F)∩F−1 ≠ ∅ if and onlyif a ∈ I. So F = ⋃a∈I C1(a). There are fifteen maximal feasible sets that arisefrom cylinders of depth 1 and all of them are regular. Moreover, whenevera ∈ I satisfies ∣a∣ ≥ 8, T [C1(a)] = F is true and we can show inductively that

(an)n≥0 ∈ IN ∶ ∀n ∈ N ∣an∣ ≥ √8 ⊆ ΩHCF

R .

The other fourteen maximal feasible are T [C1(a)] ∶ √2 ≤ ∣a∣ ≤ √5; if we

discard their boundary, we end up with thirteen regular maximal feasibleregions. Using and inductive argument and the formulae for inversion oflines and circles (see Appendix A), we conclude that these thirteen regionsare essentially all of the regular maximal feasible regions.

As an example, let us show that

T1 [C1(2)] = F ∖D(−1).If z ∈ [C1(2)], then T (z) = z−1 − 2 ∈ F, so z−1 ∈ F−1 ∩ (2 + F) and T (z) ∈F−1 ∩ (2+F) = F∖D(−1). On the other hand, when w belongs to F∖D(−1),we have that 2+w ∈ F−1∩(2+F), so (2+w)−1 ∈ C1(2) and w ∈ T [C1(2)]. Verysimilar arguments show the following equalities:

For a ∈ I with ∣a∣ = √2, a ∈ 1 + i,−1 + i,−1 − i,1 − i and

T [C1(1 + i)] = F ∖ (D(−1) ∪D(−i)) , T [C1(−1 + i)] = F ∖ (D(1) ∪D(−i)) ,T [C1(−1 − i)] = F ∖ (D(i) ∪D(1)) , T [C1(1 − i)] = F ∖ (D(−1) ∪D(i)) .

For a ∈ I with ∣a∣ = 2, a ∈ 2,2i,−2,−2iT [C1(2)] = F ∖D(−1), T [C1(2i)] = F ∖D(−i),T [C1(−2)] = F ∖D(1), T [C1(−2i)] = F ∖D(i).

8


For a ∈ I with ∣a∣ = √5, a ∈ 1+2i,2+ i,−1+2i,−2+ i,−1−2i,−2− i,1−2i,2− i

and

T [C1(1 + 2i)] = F ∖D(−1 − i), T [C1(2 + i)] = F ∖D(−1 − i),T [C1(−1 + 2i)] = T [C1(−2 + i)] = F ∖D(1 − i),

T [C1(−1 − 2i)] = F ∖D(1 + i), T [C1(−2 − i)] = F ∖D(1 + i),T [C1(1 − 2i)] = T [C1(2 − i)] = F ∖D(−1 + i).

Finally, as mentioned before, T [C1(a)] = F when a ∈ I has ∣a∣ ≥ √8. 1

0−2 −1 1 2

−2−1

1

2

Figure 1.2: The set F−1 and the lines x = k, y = l for all k, l ∈ Z.

We must deal with cylinders of level 2 to obtain an irregular maximalfeasible set. As before, we can show that for any a ∈ Z, ∣a∣ ≥ 2, we have that

(T [C1(−2))−1 ∩ (1 + ia + F) = 1 + ia + [−1 − i2

,−1 + i

2) ,

where [−1−i2 , −1+i

2 ) = z ∈ F ∶ ∃t ∈ [0,1) z = −1−i2 + ti, so

T 2[C2(−2,1 + ia)] = [−1 − i2

,−1 + i

2) .

Similarly, for a ∈ Z with ∣a∣ ≥ 2,

T 2[C2(−2,1 + ai)] =T 2[C2(−1 + i,1 − i∣a∣)] = [−1 − i2

,−1 + i

2) ,

T 2[C2(2i, a + i))] =T 2[C2(−1 + i,−∣a∣ + i)] = [−1 − i2

,1 − i

2) .

9


Proposition 1.2.1. The set of irregular numbers has Lebesgue measure 0;in fact, its Hausdorff dimension is 1.

See Chapter 4 for a longer discussion on Hausdorff dimension.

Proof. The previous discussion and the monotonicity of the Hausdorff dimen-sion give dimH FI ≥ 1. On the other hand, an inductive argument shows thatFI is contained in a countable union of arcs of circles, thus dimH FI ≤ 1.

Lack of Markovian Property

Let us study first the HCF algorithm to the real numbers. For any n ∈ Nwith n ≥ 3 we have

FR ∶= (F ∩R) ∖Q = T [[ 1

n + 1,

1

n]] = T [[− 1

n,− 1

n + 1]]

However, when n = 2 or n = −2 we have

T [[1

3,1

2]] = (0,

1

2) T [[−1

2,−1

3]] = (−1

2,0) .

Starting from these equalities— which should be read excluding rationalnumbers— we can show recursively that the associated shift space can becharacterized by the matrix

Mi,j =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0, if i = 2 and j < 0,

0, if i = −2 and j > 0,

1, in other case.

This means that an infinite sequence (an)n≥1 in Z ∖ −1,0,1 is the HCF ofsome z ∈ FR if and only if

∀n ∈ N Man,an+1 = 1.

In general, we cannot describe ΩHCF with a matrix as above. The core ofthe problem is that for a = (an)n≥1 ∈ ΩHCF and n ∈ N, the set T n[Cn(a)] doesnot only depend on an if

√5 ≤ ∣an∣ ≤ √

8. For example, while T n[Cn(a)] = Fwhenever ∣an∣ ≥ 3, we have that

T 2[C2(2 + 2i,−2 + 2i)] = F ≠ T [C1(−2 + i)] = T 2[C2(1 + 2i,−2 + 2i)].We can extend this phenomenon to arbitrarily long prefixes:

F = T 2n+2[C2n(2 + 2i,−2 + 2i, . . . ,2 + 2i,−2 + 2i)]T [C1(−2 + i)] = T 2n+2[C2n(1 + 2i,−2 + 2i,2 + 2i, . . . ,−2 + 2i,2 + 2i,−2 + 2i)].

10


|a| ≥√8

a = 2 a = −2i a = −2 a = 2i

a = 1 + i a = −1 + i a = −1− i a = 1− i

a = 2− i a = 1 + 2i a = −2 + i a = −2− i

a = 2 + i a = −1− 2i

Figure 1.3: Maximal feasible sets T [C1(a)].

In Figure 1.4 we show the sets (T [C1(a)])−1 for ∣a∣ = √2. Note that for −1+ i

the boundary makes the segments (−1+i,1−ia) and (−1+i,−a+i) admissiblefor a ∈ N, a ≥ 2. Figure 1.5 displays the sets (T [C1(a)])−1 for ∣a∣ = 2, andFigure 1.6 shows the same sets for a = 1 + 2i and a2 + i.

11


Proposition 1.2.2. There is no infinite matrix M = (Mi,j)i,j∈I such thata ∈ ΩHCF if and only if Man,an+1 = 1 for every n ∈ N.

Proof. Assume such a matrix M existed. On the one hand, the segment1+2i,−2+2i,1+ i is not admissible, so M−2+2i,1+i = 0. However, 0,−2+2i,1+ iis an admissible prefix, which would imply M−2+2i,1+i = 1, a contradiction.

0

an = 1 + i an = −1 + i an = −1− i an = 1− i

Figure 1.4: T [C1(a)]−1 for ∣a∣ =√2.

0

an = 2 an = 2i an = −2 an = −2i

Figure 1.5: T [C1(a)]−1 for ∣a∣ = 2.

1.2.2 Growth of Denominators

Take a = (an)n≥0 ∈ ΩHCF. So far, we know that the sequence denominators,(qn)n≥0, never vanishes. However, we have not said a word about its growth.The monotonicity of (∣qn∣)n≥0 goes back to A. Hurwitz in [H87]. More thana century later, Doug Hensley showed in [He06] that (∣qn∣)n≥0 in fact growsexponentially. Some years later, S.G. Dani and A. Nogueira improved in[DaNo14] D. Hensley’s result.

12


an = 1 + 2i an = 2 + i

Figure 1.6: T [C1(a)]−1 for some ∣a∣ =√5.

Theorem 1.2.3 (Hurwitz, 1887). If (qn)n≥0 is the sequence of denominatorsof a valid sequence (an)n≥0, then

1 = ∣q0∣ < ∣q1∣ < ∣q2∣ < . . . .Proof. Suppose that the result is false. Let (an)n≥0 be a valid sequencewith Q-pair ((pn)n≥0, (qn)n≥0) and let m ∈ N be minimal with respect to theproperty ∣qm∣ ≤ ∣qm−1∣, hence

∣q0∣ < ∣q1∣ < . . . < ∣qm−1∣, ∣qm−1∣ ≥ ∣qm∣.We certainly have m > 1, because 1 = ∣q0∣ < ∣q1∣, for q0 = 1 and q1 = a1 ∈ I.Define the sequence (kn)mn=1 by kn = qn/qn−1, so

∣km∣ ≤ 1 < ∣k1∣, . . . , ∣km−1∣.From ∣km−1∣ > 1 and

km = am + qm−2

qm−1

= am + 1

km−1

∈ D(am) ∩D.

we obtain that am ∈ 1+i,−1+i,−1−i,1−i. The possible regions where km canlie are shown in Figure 1.7. The four options for am are handled in a similarfashion, so let’s assume that am = 1 + i. We thus have k−1

m−1 ∈ D ∩D(−1 − i)and

km−1 = am−1 + 1

km−2

∈ E(1,1) ∩D(−1 + i).In Figure 1.8 we can see that any choice of an−1 other than am−1 = −2 + 2ileads to a feasible region for wm entailing am ≠ 1+ i. Therefore, am−1 = −2+2iand k−1

m−2 = km−1 − am ∈ D ∩D(−1 + i). An analogous argument tells us thatam−2 = 2 + 2i and we end up dealing with either of the sequences

(a0, . . . , am−1, am) = (2 + 2i,−2 + 2i, . . . ,2 + 2i,−2 + 2i,1 + i),(a0, . . . , am−1, am) = (−2 + 2i,2 + 2i, . . . ,2 + 2i,−2 + 2i,1 + i),13


depending on the parity of m.Suppose that m is even (m odd is treated similarly). Consider the se-

quence (hn)n≥1 is given by

h1 = 2 + 2i, ∀n ∈ N h2n = −2 + 2i + 1

h2n−1

, h2n+1 = 2 + 2i + 1

h2n

.

It can be shown inductively that hn = rn((−1)n+1 + i) where (rn)n≥1 is givenby r1 = 2, rn+1 = 2 − (2rn)−1, and that rn 1 + 1/√2 as n → ∞. As aconsequence, we have that

km = 1 + i + 1

hm−1

= 1 + i + −1 − i2rn

= (1 + i) (1 − 1

2rn) = (1 + i)(rn+1 − 1).

Taking absolute values we contradict the definition of m:

∣km∣ = ∣(1 + i)(rm+1 − 1)∣ > √2

1√2= 1.

14


−2 −1 1 2

−2

−1

1

2

0

−2 −1 1

−1

1

2

0

Figure 1.7: Left: Possible locations for km. Right: Possible choices for am.

0

am = 2iam ∈ −1 + 2i,−2i+ 1am = −1 + i

1 + i 1 + i1 + i

Figure 1.8: Feasible regions for wm−1 when am = −1+ i, am ∈ −1+2i,−2i+1,and am = 2i.

−1 1 2

−2

−1

1

0

Figure 1.9: Region for k−1m−1.

15


Theorem 1.2.4 (Dani, Nogueira (2011)). Let ζ = [a0;a1, a2, a3, . . .] be anelement of C ∈ Q(i) and (pn)n≥0, (qn)n≥0 its Q-pair. The following statementshold.

i. For every n ∈ N0 we have either φ∣qn∣ ≤ ∣qn+1∣ or φ∣qn+1∣ ≤ ∣qn+2∣.ii. For every n, k ∈ N we have

∣qn+k∣ > φ[ k2]∣qn∣.

The theorem is a direct conclusion of the following lemma.

Lemma 1.2.5. Let (aj)mj=0 be valid and (qn)n≥0 the associated sequence of

denominators. Suppose that for some n ∈ [1..m − 1] and some σ,√

2 < σ < φwe have ∣qn−1∣ > σ∣qn−2∣. If ∣an∣ > 2, then ∣qn∣ > σ∣qn−1∣. If ∣an∣ ≤ 2, then∣qn+1∣ > σ∣qn∣.Proof. Case ∣an∣ > 2. On the one hand, the recurrence of (qn)n≥0 gives

∣an∣ = ∣qn − qn−2

qn−1

∣ ≤ ∣ qnqn−1

∣ + ∣qn−2

qn−1

∣ ≤ ∣ qnqn−1

∣ + 1

σ.

On the other hand, since t↦ t + t−1 is increasing for t > 1,

∣an∣ ≥ √5 = 1 +√

5

2+ (1 +√

5

2)−1 ≥ σ + 1

σ.

The result follows.Case ∣an∣ ≤ 2. Suppose that an, an+1 is regular. It can be checked directly

that ∣an∣ ≤ 2 Ô⇒ R(anan+1) ≥ 2χ√2(∣an+1∣). (1.17)

Let us write rj+1 = qjqj+1 for any valid j and r = σ−1. In order to conclude that

rn+1 ∈ rD we find z ∈ C and s > 0 such that

rn+1 ∈ D( z∣z∣2 − s2,

s∣z∣2 − s2) ⊆ rD.

The hypothesis ∣qn−1∣ > σ∣qn−2∣ translates into rn−1 ∈ rD, so

an + rn−1 ∈ D(an, r) Ô⇒ rn = 1

an + rn−1

∈ D( an∣an∣2 − r2,

1∣an∣2 − r2) .

16


Writing y = (∣an∣2 − r2)−1, z = an+1 + any, s = ry we have

rn ∈ D(any, ry) Ô⇒ an+1 + rn ∈ D(an+1 + any, ry)Ô⇒ rn+1 = 1

an+1 + rn ∈ D( z∣z∣2 − s2,

1∣z∣2 − s2) . (1.18)

In order to verify the last implication we note that ∣z∣2 − s2 > 0, because from

y(∣an∣ + r) = 1∣an∣ − r < 1√2 − r < 1√

2 − 1√2

= √2 ≤ ∣an+1∣

we obtain ∣z∣ = ∣yan + an+1∣ ≥ ∣an+1∣ − y∣an∣ > yr.The rest of the proof is to show that the disc in (1.18) is contained in rD,

which happens if

∣z∣∣z∣2 − s2+ s∣z∣2 − s2

< r ⇐⇒ r(∣z∣ − s) > 1.

Setting x = y−1 = ∣an∣2 − r2, the last inequality is equivalent to ∣an∣2 < r∣an +xan+1∣ and squaring

∣an∣4 < r2(an + xan+1)(an + xan+1).Taking α = ∣anan+1∣2 and β = 2R(anan+1), we can reduce the previous equa-tion to

x2∣an+1∣2 + (β − α)x + ∣an∣2(1 − β) < 0. (1.19)

The discriminant ∆ of the quadratic is positive, because α ≥ 4, β > 0, and

∆ = (β − α)2 − 4α(1 − β) = (α + β)2 − 4α > 0.

Denote by λ− and λ+ the real roots of (1.19):

λ− = ∣an∣2((α − β) −√∆)

2α, λ+ = ∣an∣2((α − β) +√

∆)2α

.

The inequality in (1.19) holds if and only if λ− < x < λ+ or, equivalently, if∣an∣2 − λ+ < r2 < ∣an∣2 − λ− which is

∣an∣2 (α + β −√

∆

2α) < r2 < ∣an∣2 (α + β +

√∆

2α) . (1.20)

The upper bound does not require much work: r2 < 1 < ∣an∣22 = ∣an∣2 (α+β+√∆

2α ).

Using ∆ = (α + β)2 − 4α, the lower bound can be seen to be equivalent to

σ2 < 2α∣an∣2(α + β −√∆) = α + β +√

∆

2∣an∣2 .

17


We consider now two sub-cases: ∣an+1∣ ≥ 2 and ∣an+1∣ = √2.

Case ∣an+1∣ ≥ 2. Since β ≥ 0, then

α + β −√∆

2α∣an∣2 ≥ 4∣an∣2 +√16∣an∣4 − 16∣an∣22∣an∣2

= 2 + 2

√1 − 1∣an∣2 ≥ 2 +√

2 > σ2

Case ∣an+1∣ = √2. By (1.17), R(anan+1) ≥ 2 and the conclusion is gotten

verifying each case separately. For instance, an ∈ 1−i,2,−2i when an+1 = 1+iand in each case the inequality holds.

Suppose now that anan+1 is irregular. Since ∣an∣ ≤ 2, an ∈ −1 + i,−2,2iand an+1 must have a specific form:

an = −1 + i Ô⇒ ∃m ∈ Z m ≥ 2 an+1 = 1 − iman = −2 Ô⇒ ∃m ∈ Z ∣m∣ ≥ 2 an+1 = 1 + iman = 2i Ô⇒ ∃m ∈ Z ∣m∣ ≥ 2 an+1 =m + i.

Let us first take an = −2, so an+1 = 1+ im for some integer m, ∣m∣ ≥ 2. By theformula for inversion of circles,

qn+1

qn= 1 + im + 1−2 + rn−1

∈ (1 + im) +D( −2

4 − r2,

r

4 − r2) .

Hence, for some ε ∈ D we have that

qn+1

qn= (1 − 2

4 − r2) + im + ε r

4 − r2.

It can be checked that 1r + r

4−r2 < 2 whenever φ−1 < r < 2− 12 , so

1

r+ r

4 − r2< 2 < ∣(1 − 2

4 − r2) + im∣ , (1.21)

which implies ∣ qn+1qn∣ > r−1 = σ. An analogous argument clearly holds for an = 2i

and a similar one implies an = −1 + i, for which the inequality correspondingto (1.21) is

1

r+ r

2 − r2< 2 < ∣(1 − 1

2 − r2) − i(m + 1

2 − r2)∣ .

Proof of Theorem 1.2.4. Take σ < φ. The previous lemma along with aninductive argument and ∣q−1∣ = 0 < 1 = ∣q0∣ give that either ∣qn∣ > σ∣qn−1∣ or∣qn+1∣ > σ∣qn∣ hold. The theorem follows by taking σ → φ.

18


1.2.3 Quality of Approximation

A straight forward application of the Pigeon Hole Principle argument tellsus that for any ζ ∈ C′ ∶= C∖Q(i) there are infinitely many Gaussian integersp, q, q ≠ 0, such that

∣ζ − pq∣ ≤ √

21∣q∣2 . (1.22)

On the ground of Minkowski’s Second Theorem of Convex bodies, the con-stant

√2 can be reduced to 4/π. In 1925, Lester Ford showed that the least

constant valid for all complex irrationals is 3−1/2. In other words, a com-plex number ζ is irrational if and only if there are infinitely many complexrationals p/q whose approximation to ζ is O(∣q∣−2).

We would like the convergents (pn/qn)n≥0 of a complex irrational to pro-vide an approximation of order O(∣qn∣−2). We would even like to have somesort of optimality of the approximation in terms of (qn)n≥0. Richard Lakeinworked in these problems during the 1970’s. His results, which we shall nameFirst and Second Lakein Theorem, are a satisfactory answer to our inquiries.

Although both of Lakein’s Theorems are valid for every complex number,we will only state and discuss them for irrationals belonging to F. With thesesimplifications we do not lose generality and we avoid notational nuisance.

First Lakein’s Theorems

Let ζ = [0;a1, a2, . . .] with Q-pair ((pn)n≥0, (qn)n≥0) be any member of F′. Weaim to study the approximation error

ζ − pnqn

= pnζ−1n+1 + pn−1

qnζ−1n+1 + qn−1

− pnqn

= (−1)2

q2n (ζ−1

n+1 + qn−1qn

) = (−1)2

q2n (an+1 + ζn+2 + qn−1

qn) .

Let (mn(ζ))n≥0 and (Mn(ζ))n≥0 be given by

∀n ∈ N mn(ζ) = an+1 + ζn+2 + qn−1

qn, Mn(ζ) = ∣mn(ζ)∣.

DefineB ∶= infMn(ζ) ∶ ξ ∈ F′, n ∈ N.

If B happens to be a positive real number, then we would have the quadraticapproximation of HCF we long for:

∀n ∈ N ∣ξ − pnqn

∣ ≤ 1

B∣qn∣2 .The First Lakein Theorem gives us precise information about B.

19


Theorem 1.2.6 (First Lakein Theorem, 1972). B = 1.

Proof. First, we see that Mn(ζ) ≥ 1 always holds. Then, we give a pair ofsequences (ζ(j))j≥0 and (nj)j≥0 such that Mnj(ζ(j))→ 1 as j →∞.

1. Take ζ ∈ C ∖Q(i) and n ∈ N. We consider four cases depending on ∣an+1∣.§. ∣an+1∣ ≥ √

8. The triangle inequality, ζn+2 ∈ F, and ∣qn−2∣ < ∣qn−1∣ yieldMn(ζ) > 1.

§. ∣an+1∣ = 2. By symmetry, we may assume that an+1 = 2. In order to showthat mn(ζ) = 2 + ζn+2 + qn−1

qnlies outside the unit circle, note first that

2 + ζn+2 ∈ (2 + F) ∩ F−1 = (2 + F) ∖D(1), (1.23)

qn−1

qn∈ ι[D(an)] = D( an∣an∣2 − 1

,1∣an∣2 − 1

) (1.24)

(see Figure 1.10). By elementary means (like the law of cosines, for ex-ample), we can see that

inf∣z∣ ∶ z ∈ (2 + F) ∩ F−1 = ∣1 + eiπ6 ∣ > 1.9.

−1 1 2

−1

1

0

I

J

Figure 1.10: 2 + ζn+2 ∈ (2 + F) ∩ F−1.

By (1.24), we have that

∣qn−1

qn∣ ≤ ∣an∣ + 1∣an∣2 − 1

= 1∣an∣ − 1,

which is < 0.85 whenever ∣an∣ ≥ √5 and in all of these cases, we would

obtain Mn(ζ) > 1. We only have to examine what happens when ∣an∣ ∈√2,2.

20


The Laws of Succession exclude the possibilities an ∈ −1 + i,−1 − i,−2.By (1.24), R(an) ≥ 1 yields R( qn−1qn

) > 0 and R(mn(ζ)) > 1, which implies

Mn(ζ) > 1. Hence, the conclusion holds for an ∈ 1 + i,1 − i,2. Theremaining cases are −2i and 2i. Suppose that an = 2i. If Mn(ζ) ≤ 1 weretrue, we would have for some ε ∈ D that

∣2 + 2

3i + ζn+2 + 1

3ε∣ < 1 Ô⇒ ∣2 + 2

3i + ζn+2∣ < 4

3

Ô⇒ ∣2 + 2

3i∣ ≤ 4

3+ √

2

2,

which is false. The same computations work for −2i too.

§. ∣an+1∣ = √5. An argument similar to the case ∣an+1∣ = 2 does the trick.

Some care must be taken, however. The set where ζn+2 belongs can havethe form of either T [C1(2 + i)] or T [C1(2)].§. ∣an+1∣ = √

2. By symmetry, we may assume that an+1 = 1 + i. Supposefor contradiction that Mn(ζ) < 1. Then, qn−1

qn∈ D(0,1) ∩D(−1 − i) and—

as in the proof of Theorem 1.2.3— an must be −2 + 2i. Repeating theargument, we conclude the absurdity

qn−1

qn= [0;−2 + 2i,2 + 2i] ∈ C ∖Q(i).

Therefore, Mn(ζ) ≥ 1.

Putting all the cases together we obtain B ≥ 1.

2. Define the number ξ = [0;−2 + 2i,2 + 2i] (the bar means periodicity). Wecan verify directly that ∣1 + i + ξ∣ = 1. (1.25)

Define (ζ(j))n≥1 by

∀j ∈ N ζ(j) = [0;−2 + 2i,2 + 2i, . . . ,−2 + 2i,2 + 2i´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶n times (−2+2i,2+2i)

,1 + i,wj],where Rwj ≥ 2, Iwj < −2 and ∣wj ∣ ∞ when j → ∞. Then, for everyj ∈ N

m2j+1(ζ(j)) = 1 + i + 1

wn+ [0; 2 + 2i,−2 + 2i, . . . ,2 + 2i,−2 + 2i´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

n times (2+2i,−2+2i)].

Finally, in view of (1.25), M2j+1(ζ(j))→ 1 as j →∞ and B = 1.

21


Corollary 1.2.7. If ζ ∈ C′ has Q-pair ((pn)n≥0, (qn)n≥0), then

∀n ∈ N ∣ζ − pnqn

∣ < 1∣qn∣2 .Second Lakein Theorem

Let ζ be a complex number. A rational complex number p/q is a goodapproximation to ζ if

∀q′ ∈ Z[i] ∀p′ ∈ Z[i] ∣q′∣ ≤ ∣q∣ Ô⇒ ∣qz − p∣ ≤ ∣q′z − p′∣. (1.26)

and p/q is a best approximation to ζ if

∀q′ ∈ Z[i] ∀p′ ∈ Z[i] ∣q′∣ < ∣q∣ Ô⇒ ∣qz − p∣ < ∣q′z − p′∣. (1.27)

Theorem 1.2.8 (Second Lakein Theorem, 1972). Every HCF convergent ofany ζ ∈ C is a good approximation to ζ. For Lebesgue almost all ζ ∈ C everyHCF convergent is a best approximation.

Proof. Fix ζ = [0;a1, a2, . . .] ∈ F′ with Q-pair (pn)n≥0, (qn)n≥0. Since qn−1pn −qnpn−1 = (−1)n−1 holds for all n, the vectors (qn−1, pn−1), (qn, pn) always forma base of Z[i]2. Assume (p, q) ∈ Z[i]2 are given and take (s, t) ∈ Z[i]2 suchthat

(pn pn−1

qn qn−1)(s

t) = (p

q) , so (s

t) = (−1)n−1 (qn−1 −pn−1−qn pn

)(pq) .

Let n ∈ N satisfy ∣q∣ < ∣qn∣.For brevity, we write M ∶= Mn(ζ) and consider two cases: ∣t∣ > 2/M and∣t∣ ≤ 2/M .

§ Case I. ∣t∣ > 2/M . If we had ∣qnz − pn∣ ≥ ∣qz − p∣, then

2

M

1∣qnq∣ < ∣t∣∣qnq∣ = ∣pq− pnqn

∣≤ ∣pq− ζ∣ + ∣pn

qn− ζ∣

≤ ∣qnζ − pn∣∣q∣ + 1

M

1∣qn∣2 = 1

M ∣qnq∣ ( ∣qn∣ + ∣q∣∣qn∣ ) .22


and we would obtain ∣qn∣ < ∣q∣, contradicting our choice of n. Therefore,∣qnζ − pn∣ < ∣qζ − p∣ and pn/qn is a best approximation to ζ.§ Case II. ∣t∣ ≤ 2/M . In view of the First Lakein Theorem, we may

suppose that ∣t∣ ≤ 2. We aim to write (1.26) (or (1.27)) in terms of s, t. Notethat

q = sqn + tqn−1 = qn (s + tqn−1

qn) Ô⇒ ∣q∣∣qn∣ = ∣s + tqn−1

qn∣ .

Using ζ = pn−1ζ−1n+1+pnqn−1ζ−1n+1+qn we can write ζ−1

n+1 in terms of ζ as ζ−1n+1 = qnζ−pn−qn−1ζ−pn−1 .

Therefore,

∣qζ − p∣ = ∣(sqn + tqn−1)ζ − (spn + tpn−1)∣= ∣s(qnζ − pn) + t(qn−1ζ − pn−1)∣ = ∣qnζ − pn∣∣s − tζ−1n+1∣.

The pending implication is, thus,

∣s + tqn−1

qn∣ ≤ 1 & ∣t∣ < 2 Ô⇒ 1 ≤ ∣s − tzn+1∣. (1.28)

The restrictions ∣t∣ < 2 and t ∈ Z[i] imply ∣t∣ ∈ 0,1,√

2. We may discardt = 0, for it gives p = upn, q = uqn for some unit u ∈ U ∶= 1, i,−1,−i. We willcheck the two remaining options separately.

Proposition 1.2.9. Assume that ∣s + tqn+1/qn∣ ≤ 1. If ∣t∣ = 1, then ∣s∣ < 2. If∣t∣ = √2, then s ≤ 2.

Proof. The triangle inequality applied to the hypothesis gives

∣s∣ < 1 + ∣t∣.The first implication is now trivial. In order to conclude the second one, wemust show that ∣t∣ = √

2 and ∣s∣ = √5 cannot hold simultaneously. All the

possibilities are treated in a similar way, so let us show that t = 1+i, s = 1+2iis not a valid choice.

By the formula of inversion of circles,

∣s + tqn−1

qn∣ = 1√

2∣3 + i

2+ qn−1

qn∣ Ô⇒ qn−1

qn∈ D(3 + i

2,

1√2)

Ô⇒ qnqn−1

∈ D(3 − i4,

1

2√

2) ,

which gives an = 1 − i and forces n > 1. As a consequence, the previousmembership and the equality qn−2

qn−1 = −(1 − i) + qnqn−1 tell us that for some

εn−1 ∈ Dan−1 + εn−1 = qn−1

qn−2

∈ ι [−(1 − i) +D(3 − i4,

1

2√

2)] = D(−1 − 3i

2,

1√2) =∶D1.

23


The set of Gaussian integers a,√

2 ≤ ∣a∣, for which D(a) ∩D1 ≠ ∅ is

1 − i,1 − 2i,−1 − i,−2 − i,−2i,−1 − 2i,−2 − 2i,−3.(see Figure 1.11). Since an = 1 − i, we can immediately discard 1 − i,−1 −i,−1 − 2i,−2i by the laws of succession. The remaining cases are verifiedindividually. If an−1 = 1−2i, then an−2 exists and it must belong to −1−i,−2.However, the segments (−2,1− 2i) and (−1− i,1− 2i) are not valid, so an−1 =1−2i is not a valid choice. If an−1 = −2−2i, then an−2 ∈ 2,1−i, but (2,−2−2i)and (1 − i,−2 − 2i) are not valid, so an−1 ≠ −2 − 2i. Finally, an−1 = −3 forcesan−2 ∈ −1−i,−2i,1−i but (1−i,−3), (−2i,−3), and (−1−i,−3) are not valid.Hence, ∣t∣ = √

2 and ∣s∣ = √5 is impossible and ∣t∣ = √

2 implies ∣s∣ ≤ 2.

−3 −2 −1 1 2

−4

−3

−2

−1

0

Figure 1.11: Red discs are discarded directly by the Laws of Succession; bluediscs are discarded two steps. The black circle is D1.

We finish the proof by verifying the conclusion separately for ∣t∣ = √2 and∣t∣ = 1. But first, we record the next inequality, which follows from ζn+1 ∈ F,

∣ζ−1n+1∣ ≥ 2, ∀u ∈ U ∣ζ−1

n+1 − u∣ ≥ 1. (1.29)

If ∣t∣ = √2, then ∣s∣ ∈ 0,1,

√2 by Proposition 1.2.9.

i. If ∣s∣ = 0, then ∣s − tζ−1n+1∣ = ∣tζ−1

n+1∣ > 1.

ii. If ∣s∣ = 1, then ∣s − tζ−1n+1∣ ≥ 2 − 1 = 1. The equality holds if and only if

tζ−1n+1 = 2u and s = −u for some u ∈ U, which happens for only finitely

many choices of t, u, ζn+1.

24


iii. If ∣s∣ = √2, then s = ut for some u ∈ U, and

∣s − tζ−1n+1∣ = ∣ut − tζ−1

n+1∣ = ∣t∣∣u − ζ−1n+1∣ ≥ √

2 > 1.

If ∣t∣ = 1, then ∣s∣ ∈ 0,1 by Proposition 1.2.9. By the symmetries of theconstruction, we can assume that t = 1.

i. If ∣s∣ = 0, then ∣s − tζ−1n+1∣ > 1.

ii. If ∣s∣ = 1, then ∣s − tζ−1n+1∣ = ∣s − zn+1∣ ≥ 1 by (1.29). The equality happens

if and only if ζ−1n+1 = s(1 + i).

iii. If ∣s∣ = √2, we can take s = 1 + i. We argue by contradiction. Suppose

we had ∣1 + i + qn−1

qn∣ ≤ 1, ∣1 + i − ζ−1

n+1∣ < 1.

As in Theorem 1.2.3, we can obtain recursively the absurdity qn−1/qn =[0;−2 + 2i,2 + 2i] /∈ Q(i). Therefore, we must have ∣1 + i − zn+1∣ ≥ 1.

In a few words, every HCF convergent of any ζ ∈ C is a good approxima-tion of ζ. The above argument shows also that the equality in the consequentof (1.28) is satisfied only by numbers belonging to countably many circles.Thus, the almost everywhere assertion follows.

Badly Approximable Complex Numbers

A complex number is badly approximable whenever the approximation rategiven by Dirichlet’s (or Minkowski’s, or Ford’s or Lakein’s) inequality (1.22)cannot be improved significantly.

Definition 1.2.2. A complex number ζ ∈ C if badly approximable if thereis some C > 0 such that every p, q ∈ Z[i], q ≠ 0, satisfy

∣ζ − pq∣ > C∣q∣2 .

The set of badly approximable complex numbers is denoted BadC.

BadC shares some properties with its well known real counterpart. Forinstance, we will show that the Lebesgue measure of BadC is 0, and thatdimH BadC = 2. In fact, the set BadC is 1

2 -winning in the sense of SchmidtGames (see Chapter 4). Before addressing these properties, we characterizeBadC in terms of HCF. The Second Lakein Theorem allows us to extend theusual argument (See Theorem 23 in [Kh], p.36) to the complex context. Wegive it for completeness sake.

25


Lemma 1.2.10. Let ζ = [0;a1, a2, . . .] ∈ F be an irrational complex numberwith Q-pair ((pn)n≥0, (qn)n≥0), then

∀n ∈ N 1(∣ζ−1n+1∣ + 1) ∣qn∣2 < ∣ζ − pn

qn∣ ≤ 2∣qnqn+1∣ . (1.30)

Proof. Keep the notation as in the statement. The upper bound follows fromthe First Lakein First Theorem and the growth of (∣qn∣)n≥0:

∣ζ − pnqn

∣ ≤ ∣ζ − pn+1

qn+1

∣ + ∣pnqn

− pn+1

qn+1

∣ < 1∣qn+1∣2 + 1∣qnqn+1∣ ≤ 2∣qnqn+1∣ .By Proposition 1.1.1, we have for every n ∈ N that

ζ = pnζ−1n+1 + pn−1

qnζ−1n+1 + qn−1

= pnqn

+ (−1)nq2n (ζ−1

n+1 + qn−1qn

) .The lower bound follows from the growth of (∣qn∣)n≥0 and ∣ζ−1

n ∣ ≥ √2.

For our purposes, the coefficient 2 in the upper bound in (1.30) is irrel-evant as long as it is constant. With this in mind, we could even avoid theuse of the First Lakein Theorem. Indeed, keeping the above notation,

ζ − pnqn

= (−1)nqn(qnξn+1 + qn−1) = (−1)n

qn (qn+1 + qn 1ζn+2) = (−1)n

qnqn+1 (1 + qnqn+1

1ζn+2) . (1.31)

Since (∣qn∣)n≥0 is strictly increasing and ζ−1n+2 ∈ F ⊆ D(0,2− 1

2 ), we obtain

∣1 + qnqn+1

1

ζn+2

∣ ≥ 1 − √2

2> 0.

The desired bound comes now from plugging the previous inequality into(1.31).

Theorem 1.2.11. BadC = z = [a0;a1, a2, . . .] ∈ C ∶ ∃M > 0 ∀n ∈ N0 ∣an∣ ≤M .Proof. As before, we focus on F. We start with ⊇. Suppose that ζ =[0;a1, a2, . . .] ∈ F′ has elements bounded by M > 0. Then,

∀n ∈ N ∣ζ−1n ∣ = ∣an + ζn+1∣ ≤M + 1.

By the left inequality of Lemma 1.30, we have for c1 = (M + 2)−1 that

∀n ∈ N c1∣qn∣2 ≤ ∣ζ − pnqn

∣ .26


Now, take p/q ∈ Q(i) in lowest terms and n ∈ N such that ∣qn−1∣ < ∣q∣ ≤ ∣qn∣.By Theorem 1.2.8, ∣qnζ − pn∣ ≤ ∣qζ − p∣ and our choice of n gives

∣ζ − pq∣ ≥ ∣qn∣∣q∣ ∣ζ − pn

qn∣ ≥ c1∣qn∣2 > c1∣q∣2 ∣qn−1∣2∣qn∣2 = c1∣q∣2 1

∣an + qn−2qn−1 ∣2

≥ c1∣q∣2(M + 1)2.

Therefore, ζ ∈ BadC with constant C = (M + 1)−1(M + 2)−2.In order to show the other contention, take ζ ∈ BadC and let C = C(ζ) > 0

be the constant from the definition. By Lemma 1.30, for some constant c > 0and any n ∈ N we have

C∣qn∣2 ≤ c∣qnqn+1∣ Ô⇒ ∣an+1 + qn−1

qn∣ = ∣qn+1

qn∣ ≪ζ 1 Ô⇒ ∣an+1∣ ≪ζ 1.

The last implication follows from ∣qn−1∣ < ∣qn∣.Recently, another proof of Theorem 1.2.11 was given by Robert Hines

(Theorem 1 in [Hi17]). His argument also relies on R. Lakein’s work butavoids Lemma 1.30.

1.2.4 Serret’s Theorem

The complicated structure of ΩHCF is not the only difference between regular(or nearest integer) continued fractions and the Hurwitz algorithm. We shallsee that a classical theorem on the action of PGL(2,Z) over R via Mobiustransformations does not extend to HCF.

Theorem 1.2.12 (J.A. Serret, 1866). Two real numbers α = [a0;a1, . . .], β =[b0; b1, . . .] are equivalent under the action of PGL(2,Z) if and only if eitherthey are both rational or if for some j, k ∈ N we have aj+n = bk+n for all n ∈ N.

It is natural to ask for a similar result in the complex plane replacingregular continued fractions by HCF and PGL(2,Z) by PGL(2,Z[i]). Clearly,if ζ and ξ have infinte HCF expansion and their tails coincide, then they areequivalent under the action of PGL(2,Z[i]). However, R. Lakein ([La74a])proved that the converse fails. Indeed, define

Ξ = i + (43 + 28i) 12

2, A = 5 − i −Ξ

4 − i , B = 3 + 2i +Ξ

4,

the numbers A and B are equivalent under the action of PGL(2,Z[i]): A =(2B − i)/(B − i), but their HCF expansions are

A = [2 + i,3i,−1 + 2i,−1 + 2i,3,−2 − i],B = [2 + i,−2 + i,−2 + i,1 − 2i,−1 − 2i,1 + 2i].

27

1.3 Notes and Comments Basic Properties

1.3 Notes and Comments

1. Our iteration sequence notation differs from the one proposed by S.G.Dani and A. Nogueira. They use (zn)n≥0 for iteration sequences, while weuse it for the orbit of z under an f -Gauss map.

2. R. Lakein showed that Serret’s Theorem did not hold in general. However,A. Lukyanenko and J. Vandehey proved recently in [LuVa18] that Serret’sTheorem holds almost everywhere.

3. The original proof of Theorem 1.2.4 in [DaNo14] did not consider irreg-ularities. In the original exposition of Theorem 1.2.6 in [La73], the cases∣an∣ ∈ √2,

√5 are disregarded.

28

Chapter 2

Algebraic properties

In this chapter, we survey some algebraic properties of Hurwitz ContinuedFractions. We start by discussing an Euler-Lagrange-type theorem for iter-ation sequences. Afterwards, we see that Galois’ result on purely periodicregular continued fractions holds under additional restrictions and that afull analogue is impossible. Later, we give a full proof of W. Bosma’s andD. Gruenewald’s extension of D. Hensley’s work showing that there existalgebraic complex numbers of arbitrary large degree and bounded HCF ex-pansion. We end the chapter by giving combinatorial conditions on infiniteHCF (in the spirit of Y. Bugeaud) implying the transcendence of their limit.

Notation. For a given sequence (an)n≥0 in Z[i], [a0;a1, a2, . . .] representsa HCF and ⟨a0;a1, a2, . . .⟩ represents a complex continued fraction (not nec-essarily Hurwitz). If I ⊆ R is a compact interval and γ ∶ I → C is a closedpath, the winding number of γ around z is Indγ(z) ∶= (2πi)−1 ∫γ dw(w−z) . For

m,n ∈ Z we write [m..n] = j ∈ Z ∶m ≤ j ≤ n.

References. For the Euler-Lagrange-type Theorem, see [DaNo14] and [Da15].For the construction of algebraic numbers of degree greater than 2 over Q(i)and bounded HCF elements, see [He06] and [BoGu12]. For purely periodicand transcendental HCF, see [Gon18-01].

2.1 Complex Algebraic Numbers of Degree 2

A very well-known result on real continued fractions is the Euler-Lagrangecharacterization of quadratic irrationals: an irrational number is quadraticif and only if its regular (nearest integer) continued fraction expansion isperiodic. A. Hurwitz ([H87]) showed the corresponding result for quadratic

29

2.1 Complex Algebraic Numbers of Degree 2 Algebraic properties

complex numbers (over Z(i)) and his continued fractions. More than a cen-tury later, in 2011, S.G. Dani and A. Nogueira ([DaNo14]) gave a similarresult for more general complex continued fractions algorithms. S.G. Daniextended afterwards ([Da15]) the result to Eisenstein integers and their fieldof fractions.

Theorem 2.1.1 (S. G. Dani, A. Nogueira, 2011). Let z ∈ C be a quadraticsurd over Q(i). For any C > 0 there is a finite set F ⊆ C such that wheneverw = (wn)n≥0 is a non-degenerate iteration sequence of z and

N (w,C) ∶= n ∈ N ∶ ∣pn−2 − zqn−2∣ < C∣qn−2∣ , ∣pn−1 − zqn−1∣ < C∣qn−1∣is infinite, then wn ∶ n ∈ N (w,C) ⊆ F .

Proof. Let P ∈ Z[i][X] be the minimal polynomial of z, P (X) = aX2+bX+c.Take C > 0, w = (wn)n≥1, and N = N (w,C) as in the statement. In view ofProposition 1.1.1 in Chapter I, for every n ∈ N we have

a(wnpn−1 + pn−2

wnqn−1 + qn−2

)2 + b(wnpn−1 + pn−2

wnqn−1 + qn−2

) + c = 0.

Expanding and simplifying the previous expression, we obtain that

∀n ∈ N Anw2n +Bnwn +Cn = 0, (2.1)

where

An = ap2n−1 + bpn−1qn−1 + cq2

n−1,

Bn = 2apn−1pn−2 + b(pn−1qn−2 + qn−1pn−2) + 2cqn−1qn−2,

Cn = ap2n−2 + bpn−2qn−2 + cq2

n−2.

Since the roots of P are irrational, An ≠ 0 always hold. Moreover, it can beverified directly that for all n ∈ N

An = (pn−1z − qn−1)(a(pn−1 − zqn−1) + (2az + b)qn−1).The non-degeneracy hypothesis implies lim supn→∞ ∣wn∣ > 1 (cfr. Theorem1.1.2), so ∣pn−1 − zqn−1∣ → 0 as n → ∞ (cfr. Proposition 1.1.1) and for anylarge n ∣(pn−1z + qn−1)2a∣ ≤ 1.

Thus, for any large n ∈ N∣An∣ ≤ ∣(pn−1z − qn−1)qn−1(2az + b)∣ + 1 ≤ C(2az + b) + 1.

30

Algebraic properties 2.1 Complex Algebraic Numbers of Degree 2

The previous bound also holds for Cn, because Cn = An−1 and, since thediscriminant is invariant (i.e. B2

n−4AnCn = b2−4ac for all n), (Bn)n∈N is alsobounded. Therefore, the sequences (An)n∈N , (Bn)n∈N , (Cn)n∈N take finitelymany values, so the set of polynomials Pn(Z) = AnZ2 +BnZ +Cn ∶ n ∈ N,and hence the set of their roots

F = w ∈ C ∶ ∃n ∈ N Pn(w) = 0are finite. The condition wn ∶ n ∈ N ⊆ F follows from (2.1).

In general, we cannot conclude the periodicity of an iteration sequence ofa quadratic irrational. However, for certain choice functions f the associatedf -sequences are periodic. HCF are an example.

Corollary 2.1.2 (A. Hurwitz, 1897). A complex irrational z ∈ C satisfies[Q(i, z) ∶ Q(i)] = 2 if and only if its HCF is eventually periodic.

Proof. Let z ∈ C′ satisfy z = [a0; . . . , an−1, an+1, . . . , an+m] where the bar de-notes periodicity and let ((pn)n≥0, (qn)n≥0) be its Q-pair. By Proposition1.1.1 and since qn ≠ 0, we have that [an; . . . , an+m] is a quadratic irrationalover Q(i) and the same holds for z.

Now assume that z = [a0;a1, . . .] is a quadratic irrational and let (wn)n≥0

be the iteration sequence related to its HCF. By Theorem 2.1.1, for twodifferent natural numbers n,m we have wn = wm. Hence, their HCF coincide:

wn = [an;an+1, an+2, . . .], wm = [am;am+1, am+2, . . .],and (an)n≥0 is eventually periodic.

2.1.1 Purely Periodic HCF

A well known result by Evariste Galois ([Ga29]) states a real quadratic irra-tional α > 1 has a purely periodic regular continued fraction if and only if itsGalois Conjugate β satisfies −1 < β < 0. Such quadratic irrationals are calledreduced. Under some extra restrictions, we can obtain the correspondingresult for HCF.

Theorem 2.1.3. Let ξ = [a0;a1, a2, . . .] be a quadratic irrational over Q(i)and η ∈ C its Galois conjugate over Q(i).

1. If (an)n≥0 is purely periodic, then ∣η∣ < 1.

2. If ∣ξ∣ > 1, η ∈ F and ∣an∣ ≥ √8 for every n ∈ N0, then ξ has purely periodic

expansion.

31

2.1 Complex Algebraic Numbers of Degree 2 Algebraic properties

3. The conditions η ∈ F and (∀n ∈ N ∣an∣ ≥ √8) cannot be removed from the

second point. In fact, there are infinitely many pairs ξ, η such that

i. η /∈ F, ∣an∣ < √8 for some n, and (an)n≥0 is not purely periodic,

ii. η /∈ F, ∣an∣ ≥ √8 for all n, and (an)n≥0 is not purely periodic,

iii. η ∈ F, ∣an∣ < √8 for some n, and (an)n≥0 is not purely periodic.

We shall only mention the important aspects of the argument, the detailsare in [Gon18-01]. The core of the first and third points is that the Galoisconjugate (over Q(i)) η of a purely periodic HCF α = [a0;a1, . . . , am−1] canbe written as

η = − ⟨0;am−1, . . . , a1, a0⟩ . (2.2)

The mirror formula and the strict growth of the corresponding (qn)n≥0 give∣η∣ ≤ 1. The strict inequality is shown using an alternate sequence argument(similar to the one of Theorem 1.2.3. Although some new technicalities arise,most of the real argument can be used to show the second point (cfr. Theorem7.2 in [NMZ]). In order to prove the third point, we just force the continuedfraction in (2.2) to break the Laws of Succession.

There are several reasons for which the continued fraction (2.2) mightfail to be a HCF. When these reasons are simple (in some sense), we cancompute the HCF of η with a singularization1 identity like

a + 1

2 + 1

b

= a + 1 + 1

−2 + 1

b + 1

, a + 1

1 + i + 1

b

= a + (1 − i) + 1

−(1 + i) + 1

b + 1 − i.

for any a, b ∈ C. A concrete example is given by

ζ = [5 + 6i;−3 + 2i,2,9 + 4i]and its Galois Conjugate η. The sequence obtained by repeating the reversedperiod of the HCF of ζ is not valid, but using the first singularization formulawe get

η = −[0; 10 + 4I,−2,−2 + 2i,5 + 6i].Once we know that the period cannot be reversed, it is not hard to determinethe process to obtain a valid expansion. The difficulty is in determining thesource of non-reversibility.

1We borrow the term singularization from [IoKr02].

32

Algebraic properties2.2 Complex Algebraic Numbers of Degree Greater than 2

2.2 Complex Algebraic Numbers of Degree

Greater than 2

No matter how deep and extensive the theory of regular continued fractionshas become, some problems remain open. It is still unknown if 2

13 has a

bounded continued fraction expansion, for example. More generally, thefollowing conjecture has not been solved.

Conjecture 2.2.1 (Folklore Conjecture). The only badly approximable num-bers that are algebraic are the quadratic irrationals.

A strong related result was proven by Yann Bugeaud in [Bu13] on thebasis of his joint work with Boris Adamczewski in [AdBu05]. We discussbelow his result in further detail. In the mean time, it is enough to saythat they gave sufficient combinatorial conditions on the continued fractionexpansion for the transcendence of the limit.

In [He06], Doug Hensley showed what we consider to be the most impres-sive result in the Hurwitz Continued Fraction Theory: a negative answer tothe Folklore Conjecture for HCF. Hensley gave explicitly complex irrationalsz ∈ C with bounded HCF satisfying [Q(i, z) ∶ Q(i)] = 4. Some years later,in [BoGu12], W. Bosma and D. Gruenewald extended D. Hensley’s result toalgebraic numbers of arbitrary even degree over Q(i).

The proof of the result of Hensley and the later extension rely on thestudy of some circles naturally associated to each complex irrational.

2.2.1 Generalized Circles

By a generalized circle we mean the set of solutions w ∈ C of an equation

C ∶ A∣w∣2 +Bw +Bw +D = 0, (2.3)

where A,D ∈ R are given and satisfy B ∈ C, ∣B∣2 −AD ≥ 0. We represent Cby the matrix

C = (A BB D

) . (2.4)

Generalized circles are either circles or lines in the complex plane, but all ofthem are just circles in the Riemann sphere. The next properties are verifiedthrough straight forward computations.

Proposition 2.2.1. Let C be a generalized circle represented by (2.4).

33

2.2 Complex Algebraic Numbers of Degree Greater than 2Algebraic properties

i. C is a line in the plane if and only if A = 0. In this case, it is the linedetermined by

2R(wB) = −D.Otherwise, C is the circle

C = C ⎛⎝−BA,√∣B∣2 −AD∣A∣ ⎞⎠ .

ii. 0 ∈ C if and only if D = 0.

iii. IndC(0) = 0 if and only if AD > 0.

iv. IndC(0) = 1 if and only if AD < 0.

Recall that ι ∶ C∗ → C∗ is given by ι(z) = z−1. For any α ∈ C denote byτα the translation by α, τα(z) = z + α for any z ∈ C.

Proposition 2.2.2. Let C be a generalized circle given by (2.4), then theinversion of the circle is

ι[C] = (0 11 0

)(A BB D

)(0 11 0

) = (D B

B A) ,

and the translation by α,

τα[C] = ( 1 0−α 1)(A BB D

)(1 −α0 1

) = ( A −Aα +B−αA +B A∣α∣2 −Bα −Bα +D) .The previous proposition can be verified directly. Note that it gives ma-

trix representations of ι[C] and τα[C] where the determinant is invariant.

2.2.2 Circles of a complex irrational

Take a complex irrational z = [a0;a1, a2, . . .] ∈ C and define the sequence offunctions (Hn)n≥1 by Hn = τ−an ι for n ∈ N, that is

∀w ∈ C∗ Hn(w) = 1

w− an.

We also associate to z a sequence of generalized circles (Cj)j≥0 given by

C0 ∶= (A0 B0

B0 D0) ∶= ( 1 a0

a0 ∣a0∣2 − ∣z0∣2) ,∀n ∈ N Cn ∶= (An+1 Bn+1

Bn+1 Dn+1) ∶=Hn[Cn−1].

34

Algebraic properties2.2 Complex Algebraic Numbers of Degree Greater than 2

Even if their definition is somehow involved, the sequence (Cj)j≥0 arise ina natural way. To see it, define (zn)n≥1 by z0 = z − a0 and zn = T n−1(z0) =[0;an, an+1, . . .] for n ∈ N. Hence, by definition z1 ∈ C0(−a0, ∣z∣), so z2 =H1(z1) ∈H1[C0] = C1. In general, we may verify inductively that

∀n ∈ N zn+1 =Hn(zn) ∈Hn[Cn−1] = Cn.

By Proposition 2.2.2, the computation of the determinant gives

∀n ∈ N AnDn − ∣Bn∣2 = A0D0 − ∣B0∣2 = −∣z∣2. (2.5)

The main result of W. Bosma and D. Gruenewald is the following.

Theorem 2.2.3 (W. Bosma, D. Gruenewald, 2011). For any z ∈ C ∖Q(i)with ∣z∣2 = n ∈ N, the collection Cj ∶ j ≥ 0 is finite.

Proof. The finiteness is obtained by showing that the sequences of Gaussianintegers (Aj)j≥0, (Bj)j≥0, and (Cj)j≥0 take only finitely many values. Theconditions ∀j ∈ N0 ∣Bj ∣2 −AjDj = n, Cj ∩ F ≠ ∅.lie in the heart of the argument. We start working with the simpler case ofCj being a line and later we assume that it is a circle (in the plane).

§. Cj is a line. Because of Aj = 0, we have ∣Bj ∣2 = n and there are onlyfinitely many possibilities for Bj. To bound Dj, note that Cj is the line givenby

R(Bjw) = −Dj

2.

Therefore, since Cj ∩ F ≠ ∅ we can take w ∈ Cj ∩ F to obtain

∣Dj ∣2

= ∣R(Bjw)∣ ≤ ∣Bj ∣∣w∣ ≤ √2

2∣Bj ∣ <

√2n

2.

And conclude that there are only finitely many options for Dj.

§. Cj is a circle. Let Rj = √∣B∣2−AD∣A∣ be the radius of Cj. The conclusion fol-

lows from Rj > 1/√8. Indeed, in the presence of such inequality, Proposition2.2.1 would give

∣A∣ ≤ ∣BJ ∣2 −AjDj

8= n

8.

35

2.2 Complex Algebraic Numbers of Degree Greater than 2Algebraic properties

Again, by Proposition 2.2.1 and by Cj ∩ F ≠ ∅,

∣−Bj

Aj∣ ≤ √

2

2+Rj =

√2

2+ √

n

Aj,

which implies ∣Bj ∣ ≤ ∣Aj ∣(√n + 2) ≤ K1(n). Since Aj and Bj determine Dj

and both Aj and Bj can take finitely many values, the same holds for Dj.Therefore, we could conclude the finiteness of the family of circles Cj ∶ j ∈N0.

We show by induction on j that Rj > 1/√8. Since Cj is a line if Cj−1 goesthrough the origin, we consider the next three possibilities:

1. Cj−1 is a line not containing 0,

2. Cj−1 is a circle and IndCj−1(0) = 1,

3. Cj−1 is a circle and IndCj−1(0) = 0.

In the three cases we use a simple observation: the radius of Cj is the sameas the radius of ι[Cj−1]. We denote by ∣Cj ∣ = 2Rj the diameter of Cj.

1. Assume that Cj−1 is a line not containing 0. Let P ∈ Cj−1 be the nearestpoint to 0, so ∣P ∣ ≤ 1/√2. Since 0, P −1 ⊆ ι[Cj−1], we conclude that

2Rj = ∣Cj ∣ ≥ ∣P −1∣ ≥ 1√2

Ô⇒ Rj > 1√8.

2. Suppose that Cj−1 is a circle around 0. By part iv of Proposition 2.2.1and the formula for inversion in Proposition 2.2.2, we get Indι[Cj](0) = 1.Also, by zj ∈ Cj−1, z−1

j ∈ ι[Cj−1] and, calling c the center of ι[Cj−1],2Rj > ∣z−1

j − c∣ + ∣c − 0∣ ≥ ∣z−1j ∣ ≥ √

2 Ô⇒ Rj > 1√8.

3. Assume that Cj−1 is a circle and IndCj−1(0) = 0. Let P ∈ Cj−1 be the point

in Cj−1 closest to 0 and define p ∶= ∣P ∣ < 1/√2. Let Q be the antipodalpoint of P and write ∣Q∣ = p + 2Rj−1 = p + ∣Cj−1∣. Then, by the InductionHypothesis Rj−1 > 1/√8 and since t↦ t

p+t is increasing,

2Rj = ∣Cj ∣ ≥ ∣ι(P ) − ι(Q)∣≥ 1

p− 1

p + ∣Cj−1∣= ∣Cj−1∣p(p + ∣Cj−1∣) = 2Rj−1

p(p + 2Rj−1)> 1/√2

p(p + 1/√2) > p

p(p + 1/√2) > 11√2+ 1√

2

= 1√2.

36

Algebraic properties 2.3 Transcendental numbers

so Rj > 1/√8.

Corollary 2.2.4 (W. Bosma, D. Gruenewald, 2011). If z ∈ C∖Q(i) satisfies∣z∣2 = n ∈ N and x2 + y2 = n does not have integer solutions, then z has abounded HCF.

Proof. None of the circles Cj passes through the origin, because otherwisewe would have ∣aj ∣2 = ∣z0∣2 for aj ∈ Z[i], which is impossible. Therefore,

infj∈N0

minjd(0,Cj) > 0,

and (∣z−1n ∣)n≥0 stays bounded away from 0. This implies that the numbers

z−1n = [an;an+1, . . .] are bounded, so (aj)∞j=0 is bounded too.

By virtue of Corollary 2.2.4, we can construct infinitely many algebraicnumbers in BadC of degree greater than 2. We need to recall a result onelementary number theory. Namely, that for a given n ∈ N the equationn = x2 + y2 has integer solutions if and only if in the prime factorization of n

n = 2γ⎛⎝ ∏p≡1(4)p

αp⎞⎠⎛⎝ ∏

q≡3(4) qβq⎞⎠ .

all the exponents βq are even (Theorem 2.15. in [NMZ], p.55). For example,the number w = √

3 + i√8 is algebraic of degree 4 over Q(i) and, since∣w∣2 = 11, its HCF is bounded. More generally, let n ∈ N be a prime number

such that n ≡ 3 (mod 4). For any m ∈ N the numbers ζm = m√

2 + i√n − m√

4have bounded HCF elements and satisfy [Q(i, ζ),Q(i)] = 2m.

2.3 Transcendental numbers

Corollary 2.2.4, makes it particularly easy to construct transcendental num-bers with bounded HCF. For instance, just take any transcendental ζ ∈ [0,1]and consider

√ζ + i√7 − ζ. In this section, we give another way to construct

infinitely many transcendental complex numbers by imposing some combi-natorial properties on their HCF expansion.

2.3.1 Combinatorics on words

Let us recall some terminology. An alphabet A is a non-empty set. A wordin an alphabet A is a sequence, finite or infinite, taking values on A. We

37

2.3 Transcendental numbers Algebraic properties

denote the set of finite words on A by A∗ and the set of infinite words (tothe right) on A by Aω. Take a ∈ A∗ and b ∈ A∗∪Aω, we say that a = a1 . . . anis a sub-word of b = b1b2 . . ., and denote it a ≤ b, if bi+1 . . . bi+n = a1 . . . anfor some i ∈ N. The length of a word a = a1 . . . an is the number of terms itcomprises, we denote it by ∣a∣ = n. For a ∈ Aω and m,n ∈ N, m ≤ n, we writea[m..n] = amam+1 . . . an.

Let a ∈ Aω be an infinite word over a finite A. A way to measure howsophisticated is the structure of a is through the complexity function,pa(⋅,A) ∶ N→ N,

∀n ∈ N pa(n,A) = #x ∈ A∗ ∶ ∣x∣ = n, x ≤ a .Y. Bugeaud and D. H. Kim defined in [BuKi15] another useful device: therepetition exponent. Broadly speaking, it tells us how long do we have towait in average to see the first repetition of any sub-word of length n in a.

Definition 2.3.1. Let x = x1x2x3 . . . ∈ Aω. For every positive integer nconsider

r(n,x) = minm ∈ N ∶ ∃i ∈ [1..m − n] x[i..i+n−1] = x[m−n+1..m].The repetition exponent of x, repx, is

repx = lim infn→∞

r(n,x)n

.

Theorem 2.3.1 (Y. Bugeaud, 2011). Let a = (an)n≥0 be a non-periodic se-quence of natural numbers such that

repa < +∞.Then, [0;a1, a2, a3, . . .] is transcendental.

Let a be a sequence of natural numbers taking finitely many values. Un-ravelling the definition of the repetition exponent, we can show that

∀n ∈ N r(n,a) ≤ p(n,a) + n. (2.6)

Therefore, if we had p(n,a) ≪a n, the number [0;a1, a2, a3, . . .] would be ei-ther a quadratic irrational or trascendental. An important class of sequenceswhose complexity function has an at most linear growth is the family of au-tomatic sequences (Corollary 10.3.2, p. 304 in [AlSh03]). Thus, Theorem2.3.1 tells us that automatic regular continued fractions are either quadraticor transcendental.

38

Algebraic properties 2.3 Transcendental numbers

2.3.2 Complex Transcendental Numbers

Let A ≠ ∅ be a finite alphabet and a an infinite word over A. As noted in[BuKi15], the statement rep(a) < +∞ is equivalent to the existence of threesequences of finite words in A, (Wn)n≥1, (Un)n≥1, and (Vn)n≥1, such that

i. For every n the word WnUnVnUn is a prefix of a,

ii. The sequence (∣Wn∣ + ∣Vn∣)/∣Un∣ is bounded above,

iii. The sequence (∣Un∣)n≥1 is strictly increasing.

The behaviour of (Wn)n≥1 prompts two different cases:

lim infn→∞ ∣Wn∣ < +∞ ∨ lim inf

n→∞ ∣Wn∣ = +∞.In the first one, we can obtain satisfactory approximations of [0;a1, a2, . . .] bypurely periodic HCF. In the second case, we can approximate by eventuallyperiodic HCF.

The following result corresponds to Theorem 2.3.1 in the HCF context.Unfortunately, the slight strengthening of the hypothesis in the second pointleads to a much less elegant statement.

Theorem 2.3.2. Let a = (aj)j≥1 ∈ ΩHCF be non-periodic and such that

repa < +∞.Call ζ = [0;a1, a2, a3, . . .] and let (Wn)n≥0, (Un)n≥0, (Vn)n≥0 be as above.

i. If lim infn→∞ ∣Wn∣ < +∞, then ζ is transcendental.

ii. If lim infn→∞ ∣Wn∣ = +∞ and ∣an∣ ≥ √

8 for every n ∈ N, then ζ is transcen-

dental.

The lengthy proof of Theorem 2.3.2, found in [Gon18-01], follows theoriginal strategy used on Theorem 2.3.1. Accordingly, a fundamental tool isa suitable version of Schmidt’s Subspace Theorem. Wolfgang Schmidt showedin [Sch76] his Subspace Theorem for number fields. It reads as follows whenrestricted to Q(i).Theorem 2.3.3 (Schmidt’s Subspace Theorem for Number Fields). LetL1, . . . ,Lm and M1, . . . ,Mm be two sets of m linearly independent linearforms with algebraic coefficients.

39

2.3 Transcendental numbers Algebraic properties

For any ε > 0 there exists T1, . . . , Tk proper subspaces of Q(i)m such thatfor every β = (b1, . . . , bk) ∈ Z[i], β ≠ 0,

∣ m∏j=1

Lj(β)∣ ∣ m∏j=1

Mj(β)∣ ≤ 1∥β∥ε∞ Ô⇒ β ∈ k⋃j=1

Tj,

where ∥β∥∞ = max∣b1∣, . . . , ∣bk∣, and β = (b1, . . . , bm).

Examples of Complex Transcendental Numbers

In order to use Theorem 2.3.2, we recall some notions related to k-uniformmorphisms. Let A be an alphabet. The set A∗ along with the concatenationoperation has a monoid structure, it is called the free monoid over A. Bya morphism mean a monoid morphism ϕ ∶ A∗ → B∗, where B is anotheralphabet. Clearly, such an ϕ is determined by its action on each a ∈ A.For any k ∈ N, ϕ is k-uniform if ∣ϕ(a)∣ = k for all a ∈ A, a coding is a1-uniform morphism. A morphism is uniform if it is k-uniform for some k.A k-uniform morphism ϕ ∶ A∗ → A∗ is prolongable on a ∈ A if ϕ(a) = axfor some x ∈ A∗, and the fixed point of ϕ based at a is the infinite word

axϕ(x)ϕ2(x)ϕ3(x)ϕ4(x) . . . ,where, naturally, ϕ1 = ϕ and ϕn+1 = ϕ ϕn.

We can give right away two famous examples. Set A = a,b, where a, bare two different symbols. Let µ ∶ A∗ → A∗ be the morphism determined byµ(a) = ab, µ(b) = ba. The Thue-Morse sequence on A, t, is the fixed point(based on a) of µ:

t = abbabaabbaababba . . . .

Now let ϕ ∶ A∗ → A∗ be the morphism determined by ϕ(a) = aab, ϕ(b) = bba.The Mephisto Waltz sequence A, s, is the fixed point (based on a) of ϕ:

s = aabaabbbaaabaabbba . . . .

The following proposition is an immediate conclusion— almost a mere restatement—of well-known results by A. Cobham (see Theorem 6.8.2, p.190, Theorem6.3.2, p. 175, and Corollary 10.3.2, p.304, in [AlSh03]).

Proposition 2.3.4 (A. Cobham, 1972). Let A be a finite alphabet and a ∈A∗. Assume that one of the following the conditions hold:

1. If a is the fixed point of a uniform morphism,

2. There exist a pair of positive integers k,m such that for all r ∈ [0..m−1]the sequence (anm+r)n≥0 is the image under a coding of the fixed pointof some k-uniform morphism.

Then, rep(a) < +∞.

40

Algebraic properties 2.4 Further results

Regular transcendental numbers. Let a, b ∈ Z[i], a ≠ b, be such that∣a∣, ∣b∣ ≥ √5 and let t be the Thue-Morse sequence on a, b starting at a.

Since t is not periodic (cfr. Theorem 1.6.1., p.15., in [AlSh03]), Theorem2.3.2 tells us that ζ = [0; t1, t2, t3, . . .] is transcendental.

Irregular transcendental numbers. Unlike in the regular case, we mustbe careful while constructing an irregular sequence a satisfying rep(a) < +∞.Not every fixed point of a k-uniform morphism over a suitable alphabet willdo the trick (or, more generally, not any k-automatic sequence will work).We must keep in mind the following observation.

Proposition. Let (an)n≥1 ∈ ΩHCFI , then lim infn ∣an∣ ≤ 2. In fact, one of the

following two inequalities must hold

lim supn→∞ ∣a2n∣ ≤ 2, ∨ lim sup

n→∞ ∣a2n+1∣ ≤ 2.

Take two different rational integers m,n with ∣m∣, ∣n∣ ≥ 3 and let s =(sj)j≥1 be the corresponding Mephisto Waltz sequence. Define the worda = a1a2a3 . . . by ∀n ∈ N a2n−1 = −2, a2n = sn.Since s is not periodic (cfr. Exercise 16, p.25, in [AlSh03]), a is not periodiceither. Moreover, the sequences (Wn)n≥1, (Vn)n≥1 as in Theorem 2.3.2 canbe taken to be Wn = Vn = ε for all n. Hence, a ∈ ΩHCF

I and by Proposition2.3.2 and Theorem 2.3.2, [0;a1, a2, a3, . . .] is transcendental.

2.4 Further results

Other transcendence results can be translated into our context with the per-tinent modifications. An example is Theorem 2.4.1 (cfr. [Bu15], Theorem11.2).

We say that an infinite word a ∈ A satisfies condition (♣) if there aresequences (Wn)n≥1, (Un)n≥1, (Vn)n≥1 of finite words in A such that

1. WnUnVnUn is a prefix of a for every n ∈ N, where Un is the wordobtained by reversing Un,

2. ((∣Wn∣ + ∣Vn∣)/∣Un∣)n≥1 is bounded,

3. (∣Un∣)n≥1 tends to infinity when n does.

Theorem 2.4.1. Let a ∈ ΩHCF satisfy Condition (♣) and ∣an∣ ≥ √5 for every

n. If a is not periodic, then ζ = [0;a1, a2, . . .] is transcendental.

41

2.5 Notes and Comments Algebraic properties


1. The restriction ∣an∣ ≥ √5 in Theorems 2.3.2 and 2.4.1 are needed for

the same reason. In R, if the regular continued fractions of α and βare bounded, say by M , then ∣α − β∣ ≍m,M 1 where m is the length ofthe longest prefix of α which is also a prefix of β. This is no longertrue for HCF in general. However, if F = ⋃a C1(a), where a runs alongthe set w ∈ Z[i] ∶ ∣w∣ ≥ √

5. There is some positive ε > 0 such thatd(b + F, c + F ) > ε for any b, c ∈ Z[i] (see Figure 1.1, Chapter 1.).

2. We avoided a discussion on k-automatic sequences, because it is farsimpler to introduce the notion of fixed points of uniform morphisms.However, the following is an immediate consequence of Theorem 2.3.2.

Corollary. Let A ⊆ Z[i] be a finite set such that min∣a∣ ∶ a ∈ A ≥√5. Then, any k-automatic sequence (an)n≥1 over A is the HCF of a

transcendental complex number.

Our omission is irrelevant in view of Cobham’s characterization of au-tomatic sequences as the image under codings of fixed points of uniformmorphisms.

42

Chapter 3

Ergodic Theory of HurwitzContinued Fractions

In this chapter, we present basic results on the ergodic theory of the dy-namical system (F, T ). In particular, we show the existence of a measure µequivalent to the Lebesgue measure making (F, T, µ) ergodic. We mentionsome consequences of Birkhoff’s Ergodic Theorem including a formula forestimating the entropy of (F, T, µ), and an application on binary forms. Wefinish the chapter by mentioning two general frameworks containing HCF.One of them has been treated by F. Schweiger, H. Nakada, among others,and in a slightly modified version by A. Berechet. The second one is treatedin a recent paper by A. Lukyanenko and J. Vandehey.

Notation. Whenever the Vinogradov symbol ≪ and in ≍ appears withoutany parameters, the implied constants are absolute. The σ-algebra of Borelsubsets of F (resp. Fn(a)) is denoted by B (resp. BFn(a)). The n-th termof the HCF of a number z regarded as a function of z is denoted an(z) andsimilarly for (pn)n≥0, (qn)n≥0. If Ω is a set of sequences in Z[i] and m ∈ N,

Ω(m) ∶= (aj)mj=1 ∈ Z[i]m ∶ ∃(bj)j≥1 ∈ Ω b1 = a1, . . . , bn = an .The Lebesgue measure on F is m. If (X,M, µ) is a measure space, ∀µ x ∈ Xmeans for µ-almost all x ∈X.

Main references. For general ergodic theory see [CFS82]. For the ergodictheory of HCF and larger settings including them see [Na76], [Sc00b], [Be01],and [NaNa03]. For Iwasawa Continued fractions see [LuVa18].

43

3.1 Geometric observations Ergodic Theory of Hurwitz Continued Fractions

3.1 Geometric observations

We start by recalling a classical measure theoretic result. Its proof can befound in almost any measure theory book, e.g. [Bo07], Vol. 1., Theorem3.7.1., p. 194.

Theorem 3.1.1 (Change of Variable). Let U ⊆ Rn be open and let φ ∶ U → Rn

be differentiable with derivative Dφ. If φ is injective, then for any measurableA ⊆ U and any Borel function f ∈ L1(Rn) we have

∫φ[A] f(x)dx = ∫A(f φ)(t) ∣detDφ(t)∣dt.

The present and the next chapter rely heavily on a geometrical featureof the construction; loosely stated: cylinders look a lot like maximal feasibleregions. This is rather unsurprising considering that T n restricted to anyregular cylinder Cn(a) is bi-holomorphic. Let us elaborate in this aspect.

We adopt the following notation:

∀a ∈ ΩHCFR ∀n ∈ N Fn(a) ∶= T n[Cn(a)],

and extend it in the obvious way to finite sequences.Take n ∈ N and a ∈ ΩHCF

R with Q-pair ((pm)m≥0, (qm)m≥0). The functionT n ∶ Cn(a)→ Fn(a) is bijective, its inverse ta,n is given by

∀z ∈ Fn(a) ta,n(z) = pn−1z + pnqn−1z + qn = pn−1

qn−1

+ (−1)nqn−1(qn−1z + qn) .

Since ∣qn∣ < ∣qn+1∣, ta,n is holomorphic on its domain and

∀z ∈ Fn(a) t′a,n(z) = (−1)n+1

(qn−1z + qn)2.

Recall that if f ∶ U ⊆ C → C is holomorphic, then it is real differentiableand ∣Df(x, y)∣ = ∣f ′(x + iy)∣2 for every z = x + iy ∈ U (see [Lan99], p.33). Inparticular, for ta,n we have

∀z = x + iy ∈ Fn(a) ∣detDta,n(x, y)∣ = 1∣qn−1z + qn∣4 = 1∣qn∣4∣ qn−1qnz + 1∣4 .

As a consequence, we have the bounds

∀z ∈ Fn(a) 2−4 1∣qn∣4 ≤ ∣detDta,n(x, y)∣ ≤ (1 − √2

2)−1

1∣qn∣4 ;

44

Ergodic Theory of Hurwitz Continued Fractions 3.1 Geometric observations

or more briefly,

∣detDta,n∣ ≍ 1∣qn(a)∣4 . (3.1)

The previous expression is known as a Renyi condition.Since there are only finitely many regular maximal feasible regions (thir-

teen if we discard null sets), the Theorem of Change of Variable and (3.1).

∀B ∈BFn(a) m(ta,n[B ∩ Fn(a)]) ≍ m(B ∩ Fn(a))∣qn∣4 . (3.2)

A regular cylinder Cn(a) is full if Fn(a) = F. Computing the proportionof F that is occupied by full cylinders of depth 1 (see Figure 1.1 in Chapter I)is an exercise in Riemann integration. There are no additional complicationsif we replace F by any regular maximal feasible region. Since the restrictionsof T n to regular cylinders do not entail too much distortion (cfr. (3.1)), itis reasonable to believe that the proportion of a regular cylinder of depth noccupied by full cylinders of depth n+k can be estimated. This phenomenonis precisely stated in Lemma 3.1.2.

We remind the reader that I = Z[i] ∖ 0,1, i,−1,−i and write

∀n ∈ N A(n) = a ∈ In ∶ T n[Cn(a)] = F .Given E ⊆ F, a ∈ ΩHCF

R and m,k ∈ N, set

Υ(E,m) ∶= b ∈ Im ∶ Tm[Cm(b) ∩E] = F,Υ(a, k,m) ∶= b ∈ Im ∶ a1 . . . akb ∈ A(m + k) .

Lemma 3.1.2. For all m,k ∈ N and any a ∈ ΩHCFR (k)

∑b∈Υ(a,k,m)m (Cm+k(ab)) ≫ m (Ck(a)) . (3.3)

Proof. Let F′ ≠ ∅ be the interior of a regular maximal feasible set, the abovediscussion gives ∑

b∈Υ(F′,1)m (C1(b)) ≫ m(F′). (3.4)

For any a ∈ ΩHCFR with Q-pair ((pn)n≥0, (qn)n≥0) and k ∈ N, the estimate (3.2)

translates into

m (Ck(a)) ≍ m (Fk(a))∣qk∣4 , ∀b ∈ Υ(a, k,1) m (Ck+1(ab)) ≍ m (C1(b))∣qk∣4 .

These bounds along with (3.4) yield

∑b∈Υ(a,k,1)m (Ck+1(ab)) ≫ m (Ck(a)) . (3.5)

45

3.2 An ergodic measure Ergodic Theory of Hurwitz Continued Fractions

Take m ∈ N, m ≥ 2, and put B = b′ ∈ Im−1 ∶ Cm−1(b′) ∩ F′ ≠ ∅, then

∑b∈Υ(F′,m)m (Cm(b)) = ∑

b′∈B ∑b∈Υ(b′,m−1,1)m (Cm(b′b))

≫ ∑b′∈Bm (Cm−1(b′)) ≥ m(F′). (3.6)

When a ∈ ΩHCFR , m,k ∈ N are given, we can obtain (3.3) from (3.6) just

as we got (3.5) from (3.4).

3.2 An ergodic measure

We devote this section to construct an analogue of the Gauss measure forregular continued fractions. Unfortunately, the proof says nothing about theexact form of its density.

Recall that given a measure space (X,S, λ) and a measurable map τ ∶X → X, we say that λ is τ -invariant if λ(E) = λ(τ−1[E]) for all E ∈ S.The measure λ is τ-irreducible (or just irreducible if τ has been fixed) ifτ−1[E] = E yields λ(E) = 0 or λ(E) = 1 for any measurable E. We say thatλ is ergodic (with respect to τ) if it is τ -invariant and τ -irreducible. We saythat τ is non-singular if λ(τ−1[E]) = 0 implies λ(E) = 0 for all E ∈S.

Theorem 3.2.1 (H. Nakada, 1976). There is a unique Borel measure µ whichis equivalent to m and such that the system (F, T, µ) is ergodic.

We will require two previous results. The first one— due to N. Dunfordand D.S. Miller— is shown in [DuMi46] as Theorem 1. The proof of thesecond one can be found in V. Bogachev’s measure theory book ([Bo07], Vol.1., Theorem 4.6.3., p. 275).

Lemma 3.2.2 (N. Dunford, Miller 1946). Let (X,S, ν) be a measure spacesuch that ν(X) < +∞ and let ϕ ∶ X → X be non-singular. If there is someM > 0 such that

∀A ∈S ∀n ∈ N 1

n

n−1∑r=0

ν(ϕr[A]) ≤Mν(A),then, for all f ∈ L1(X,ν) the limit

limn→∞

1

n

n−1∑r=0

f(ϕrt).exists ν-almost everywhere in X.

46

Ergodic Theory of Hurwitz Continued Fractions 3.2 An ergodic measure

Lemma 3.2.3. Let (X,S) be a measurable space and (µn)n≥1 a sequence ofmeasures on X such that the following limits exist

∀E ∈S µ(E) ∶= limn→∞µn(E).

If µ(X) < +∞, then µ is a measure on (X,S).

Proof of Theorem 3.2.1. To construct µ, we first show that for some absoluteconstants ∀k ∈ N ∀E ∈B m(E) ≍ m (T −k[E]) . (3.7)

Lemma 3.2.2 applied to the functions χEE∈B along with Lemma 3.2.3 tellus that we can define a T -invariant measure µ via

∀E ∈B µ(E) ∶= limn→∞

1

n

n−1∑j=0

m(T −j[E]). (3.8)

By (3.7), µ and m are equivalent and the ergodicity of µ would follow fromthe irreducibility of m. We conclude that m is irreducible by showing thatany measurable E with m(E) > 0 satisfies

T −1[E] = E Ô⇒ ∀F ∈B m(E ∩ F ) ≫ m(E) m(F ). (3.9)

Taking F = F ∖ E we conclude m(F ∖ E) = 0. The uniqueness of µ followsfrom Birkhoff’s Ergodic Theorem (see Appendix A).

Proving Theorem 3.2.1 amounts now to show (3.7) and (3.9).

Proof of (3.7). For every k ∈ N define

ck ∶= minm(Fk(a)) ∶ k ∈ N, a ∈ ΩHCFR (k) .

Since m (Ck(a)) ≍ mFk(a)∣qk(a)∣−4 for any a ∈ IN,

∑a∈ΩHCF

R (k)ck∣qn(a)∣4 ≪ ∑

a∈ΩHCFR (k)

m (Ck(a)) = mF = 1

By the Theorem of Change of Variable, for all a ∈ ΩHCFR and any k

m(T −k[E] ∩ Ck(a)) = m(ta,k[E ∩ Fk(a)])= ∫

Fk(a) χE ∣detDta,k∣dm≍ m(E ∩ Fk(a)) m (Ck(a)) . (3.10)

47


In particular, whenever a ∈ A(k),m(T −k[E] ∩ Ck(a)) ≍ m(E) m (Ck(a)) . (3.11)

In view of Lemma 3.1.2, (3.11) yields

m (T −k[E]) = ∑a∈Ikm(T −k[E] ∩ Ck(a))

≥ ∑a∈A(k)m(T −k[E] ∩ Ck(a)) ≍ m(E) ∑

a∈A(k)m (Ck(a)) ≫ m(E).The other inequality also follows directly from (3.10):

∀k ∈ N m (T −k[E]) =∑a

m(T −k[E] ∩ Ck(a))≍∑

a

m(E ∩ F(a))m (Ck(a)) ≤ m(E),where a runs along ΩHCF

R (k).Proof of (3.9) Since the family of cylinders generate the Borel σ-algebra,it suffices to establish (3.9) for cylinders. Thus, we must show that for agiven set E ∈ B such that T −1[E] = E and mE > 0 the following propertyholds:

∀a ∈ ΩHCFR ∀m ∈ N m(E ∩ Cm(a)) ≫ m(E) m (Cm(a)) . (3.12)

Take a ∈ ΩHCFR (m) and m ∈ N. By the Theorem of Change of Variable

and noting that the T -invariance of E yields χE tc,k = χE for any c ∈ A(k)we have

m(E ∩ Cm(a)) =∑b∈I ∫Cm+1(ab) χE dm

≥ ∑b∈Υ(a,1)∫Cm+1(ab) χE dm

= ∑b∈Υ(a,1)∫F

(χE ta,b) ∣detDta,b∣dm= ∑b∈Υ(a,1)∫E ∣detDta,b∣dm

≫ ∑b∈Υ(a,1)

m(E)∣qm+1(ab)∣4≫ m(E) ∑

b∈Υ(a,1)m (Cm+1(ab)) ≫ m(E) m(Cm(a)).

48


It is well known that the density function of the real Gauss measure isgiven by f ∶ [0,1]→ R, f(x) = (2 logx)−1. As today, it is still unknown if thedensity function of µ with respect to m has a simple explicit form. However,using elements of thermodynamic formalism, Doug Hensley could show thatρ is real analytic almost everywhere on F.

Theorem 3.2.4 (D. Hensley, 2006). Let µ be the measure from Theorem3.2.1. The density of µ, h, is real analytic in the twelve regions determinedby the boundaries of the regular maximal feasible regions (see Figure 3.1).Moreover, the symmetries µ(iA) = µ(A) and µ(A) = µ(A) hold for everyBorel set A ⊆ F.

Figure 3.1: The sets of continuity of ρ.

We delay the discussion on the mixing properties of T . We content our-selves for the moment to state a theorem that is an immediate conclusion ofthe Nakada - Natsui Theorem discussed below.

Theorem 3.2.5. The map T is strong mixing with respect to µ; that is

∀A,B ∈B limn→∞µ(T −n[A] ∩B) = µ(A)µ(B).

3.2.1 Consequences and related results

Let us start with two natural observations providing a non-trivial differenceand a similarity between regular and Hurwitz continued fractions. Bothresults follow from Birkhoff’s Ergodic Theorem.

Proposition 3.2.6. i. There are constants K,h such that

∀mz = [a0;a1, a2, . . .] ∈ F limn→∞

1

n

n−1∑j=0

aj = K, limn→∞

1

n

n−1∑j=0

∣aj ∣ = h.

49


ii. There exists a constant c such that

∀mz = [0;a1, a2, . . .] ∈ F limn→∞ ∣qn∣ 1n = c;

in fact, c = exp(− ∫F log ∣z∣dµ).

For regular continued fractions, the first limit is infinite almost everywhereand a1 is not integrable. The second part is the analogue of the Khinchin-Levy Constant.

Proof. i. In virtue of Birkhoff Ergodic Theorem, it suffices to show thata1(z) ∶ F→ I (the first element of the HCF) belongs to L1(µ). By (3.1),we have that

∀a ∈ I ∫C1(a) ∣a1(z)∣dµ = ∫C1(a) ∣a∣dµ ≍ 1∣a∣3 ;

as a consequence, summing over I and writing ∥a∥∞ = max∣Ra∣, ∣Ia∣,

∫F∣a1∣dµ ≍∑

a∈I1∣a∣3 =∑

j≥2

∑∥a∥∞=j1∣a∣3 +

√8 ≤ 8∑

j≥2

1

j2+√

8 ≤ 8 +√8

and a1 ∈ L1(µ). The constants

h = ∫ a1dµ, K = ∫ ∣a1∣dµ,give the desired conclusion.

ii. Let z = [0;a1, a2, a3, . . .] ∈ F be irrational. Since the convergents of z arein minimal expression, the identity

pn(z)qn(z) = 1

a1 + [0;a2, a3, . . .] =1

a1 + pn−1(Tz)qn−1(Tz)

= qn−1(Tz)anpn−1(Tz) + qn−1(Tz)

implies that pn(z) = unqn−1(T (z)) for some unit un ∈ Z[i] and any n.Then, we have that

∀n ∈ N ∣pn(z)qn(z) pn−1(Tz)

qn−1(Tz)⋯p1(T n−1z)q1(T n−1z) ∣ = 1∣qn(z)∣ .

As a consequence, for all n ∈ N− 1

nlog ∣qn(z)∣ = 1

n

n−1∑k=0

log ∣pn−kqn−k (T kz)∣

= ( 1

n

n−1∑j=0

log ∣T jz∣) + 1

n(n−1∑k=0

log ∣pn−kqn−k (T kz)∣ − log ∣T kz∣) .

(3.13)

50


In view of (3.7), the density of µ, h, satisfies 0 < ess inf h and ess suph <+∞. Let ε > 0. Then, changing into polar coordinates and recalling thatr log r → 0 as r → 0, we get

∫F∖εD ∣ log ∣z∣∣dµ = ∫

F∖εD ∣ log ∣z∣∣hdm = ∫F∖εD r∣ log r∣h(r, θ)dm(r, θ) ≪ 1.

By The Monotone Convergence Theorem, z ↦ log ∣z∣ belongs to L1(µ)and, by Birkhoff’s Ergodic Theorem, the first term in (3.13) tends to∫ log ∣z∣dµ.

For the second term, let us observe that any irrational w ∈ F with Q-pair((pn)n≥0, (qn)n≥0) satisfies for any n ∈ NRRRRRRRRRRRRRR∣w∣∣pnqn ∣ − 1

RRRRRRRRRRRRRR≤ ∣ wpn

qn

− 1∣ = ∣qn∣∣pn∣ ∣w − pnqn

∣ ≪ 1∣pnqn∣ = 1∣qn−1qn∣ ≤ 1

φn.

We have used First Lakein’s Theorem and Theorem 1.2.4 of Chapter 1.

The Mean Value Theorem tells us that

∀n ∈ N ∣log ∣w∣ − log ∣pnqn

∣∣ ≪ 1

φn.

Hence, the co-factor of 1n in the second term of (3.13) is bounded and

the quotient tends to 0 as n grows.

Entropy of the HCF system

In the course of proving Proposition 3.2.6 we established that f ∶ F → R,f(z) = log ∣z∣, is integrable. The real number ∥f∥1,µ = ∫ ∣f ∣dµ is stronglyrelated to the entropy of the HCF dynamical system.

Theorem 3.2.7. Let f ∶ F → R be given by f(z) = log ∣z∣. The entropy of(F,B, µ, T ) ishµ(T ) = 4∥f∥1,µ.

Proof. Let P be the partition formed by the cylinders of depth 1, P ∶=C1(b)b∈I . The construction of µ guarantees the existence of two constantsc1, c2 such that

∀b ∈ I logm (C1(b)) + c1 ≤ logµ (C1(b)) ≤ logm (C1(b)) + c2.

And, by (3.2), there are absolute constants c′1, c′2 for which

4 log ∣b∣ + c′1 ≤ − logµ (C1(b)) ≤ 4 log ∣b∣ + c′2,51


holds for all b ∈ I. Adding over I we obtain the entropy of PHµ(P) = −∑

b∈I µ (C1(b)) logµ (C1(b)) ≪∑b∈I

log ∣b∣∣b∣4 + 1∣b∣4 ≪∑b∈I

1∣b∣3 < +∞.The Shannon-McMillan-Breiman Theorem yields

∀µ z = [0;a1, a2, . . .] ∈ F limn→∞

1

nlogµ (Cn(a)) = hµ(T );

and in light of (3.7) we can replace µ by m in the previous limit. Finally, bythe the second part of Proposition 3.2.6, for m-almost any z ∈ F we have

− 1

nlogm (Cn(a)) = log ∣qn(z)∣

n+ 1

nO(1)ÐÐ→

n→∞ ∥f∥1,µ.

Complex quadratic forms in two variables

We say that a sequence a ∈ ΩHCFR is HCF-normal if every regular prefix

is a sub-block of a. A consequence of the ergodicity of (F,B, T, µ) is thefollowing corollary pointed out in [DaNo14].

Corollary 3.2.8. With respect to m, almost every complex number is HCF-normal.

Proof. Let b ∈ Im be a regular prefix. By Birkhoff’s Ergodic Theorem,

∀µ z ∈ F limn→∞

1

n

n−1∑j=0

χCm(b)(T j(z)) = µ (Cm(b)) > 0.

Since there are countably many regular prefixes and µ is equivalent to m,every regular prefix appears in the HCF expansion of µ-almost every complexnumber z.

For any pair of complex numbers ζ, ξ ∈ C define the quadratic form Qζ,ξ ∶C2 → C by ∀(w, z) ∈ C2 Qζ,ξ(w, z) = (w − ζ)(z − ξ).On the basis of the work of S. G. Dani and A. Nogueira, we can obtain thenext theorem.

Theorem 3.2.9. For almost every pair ζ, ξ ∈ C, the set Qζ,ξ[Z[i]2] is densein C.

52


This theorem is the complex analogue of an earlier result also obtainedby S.G. Dani and A. Nogueira too.

Theorem 3.2.10. For almost every pair α1, α2 ∈ R, the set Qα,β[N2] is densein R.

The complete proofs of Theorems 3.2.9 and 3.2.10 can be found, respec-tively, in [DaNo14] as Corollary 8.8. and in [DaNo02] as Corollary 5.3..

The Gauss-Kuzmin Theorem

The genesis of metrical diophantine approximation can be traced back toa letter from C. F. Gauss to P. S. Laplace where he claimed (in modernnotation) that

∀x ∈ (0,1) limn→∞mR (T −n

R [[0, x)) = ln(1 + x)ln 2

= µR([0, x]),where TR is the Gauss map. Gauss did not publish any proof. It was until1928 when it was finally shown, by R. Kuzmin and a year later by P. Levy,that Gauss’ claim was indeed true. A detailed proof (in English) of Kuzmin’sTheorem is in [Kh]. The error term obtained by P. Levy and later by E.Wirsing (based on the work of P.Szusz) improves Kuzmin’s.

Theorem 3.2.11 (R. Kuzmin, 1928). Let (fn)n≥0 be a sequence of real-valuedand integrable functions on (0,1). Suppose that

∀n ∈ N0 fn+1(x) = ∞∑k=1

1(k + x)2fn ( 1(k + x)) .

Assume the existence of M,µ > 0 such that

∀x ∈ (0,1) 0 < f0(x) <M, ∣f ′0(x)∣ < µ.Then, there are constants λ > 0, A = A(M,µ) > 0 and a function θ, ∣θ∣ < 1such that

∀n ∈ N ∀x ∈ (0,1) fn(x) = ∥f0∥1

1 + x + θ(x)Ae−λ√n.A more modern approach to Theorem 3.2.11 requires the notion of trans-

fer operator.

53


Definition 3.2.1. Let (X,S, ν) a probability space and τ ∶ X → X measur-able non-singular map. The transfer operator or Perron - Frobeniusoperator of τ ,

Pτ ∶ L1(ν)→ L1(ν), f ↦ Pτf,

assigns to each f ∈ L1(µ) the Radon - Nikodym derivative of fd(ν τ−1) withrespect to ν. That is, Pτf ∈ L1(ν) is defined by the condition

∀E ∈S ∫EPτfdν = ∫

τ−1[E] fdν.

Remark. The non-singularity of τ is needed to ensure the absolute continuityof the signed measure E ↦ ∫τ−1[E] fdν with respect to ν (enabling the use of

the Radon-Nikodym Theorem).

Remark. In general, the transfer operator is linear and positive.

Adopt for a moment the notation of Definition 3.2.1. It is an exercise inmeasure theory to show for any f ∈ L1(µ) that Pτf is the only element ofL1(µ) verifying

∀ϕ ∈ L∞(µ) ∫ ϕPτfdµ = ∫ (ϕ τ)fdµ.

In particular, for the Gauss map in [0,1], TR, with the Lebesgue measure wehave ∀f ∈ L1(m) PTRf(x) =∑

n≥1

1(n + x)2f ( 1

n + x)(the equality should be read m-almost everywhere). Theorem 3.2.11 is, thus,a result on the exponential convergence of the iterations of the transfer op-erators on certain elements of L1(m).

The Kuzmin-type theorem we will state for HCF is restricted to a class offunctions strictly smaller than L1(m). Let H ∶= H1, . . . ,H12 be the twelveregions determined by the boundaries of regular maximal feasible regions.Let L(H) be the space of functions from F to R>0 satisfying

∃m,M > 0 m ≤ ess inf f ≤ ess sup f ≤M,∀j ∈ [1..12] f ∣Hj

is Lipschitz.

For n ∈ N set σ(n) = sup∣Cn(a)∣ ∶ a ∈ ΩHCFR (n).

Theorem 3.2.12 (F. Schweiger, 2000). There is an absolute constant 0 <α < 1 such that

∀f ∈ L(F) ∀n ∈ N ∥P nT f − h∫

Ffdm∥∞ ≪f α

√n + σ[√n].

54

Ergodic Theory of Hurwitz Continued Fractions 3.3 General frameworks

The full proof of Theorem 3.2.12 can be found in [Sc00b]. We will contentourselves to point out some important features of the argument. First, notethat Schweiger’s Theorem would follow if we had the uniform convergenceon the closure of each bounded set Hi.

Besides restricting the study to the aforementioned compact sets, thefollowing proposition is also required.

Proposition 3.2.13. A positive and almost everywhere integrable functionf satisfies PTh if and only if f = αh for some α ∈ R>0.

Proof. By the definition of the transfer operator and the T -invariance of µthat PTh = h. Now, let f ∈ L1(µ) be an almost everywhere positive such thatPhf = f and assume that ∫ fdm = 1. Then, the measure νf(E) ∶= ∫E fdm isT -invariant and, since m is irreducible, it is also ergodic. Since ν ≪ µ andboth are ergodic, we have ν = µ and f = h almost everywhere.

Take any f ∈ L(H). Clearly, f ≍f h and, by the positivity of the trans-fer operator, P n

T f ≍f P nT h = h holds for all n. Let H be the closure of some

Hi ∈ H. Suppose, without loss of generality, that f is continuous on H. Then,from n−1∑n−1

j=0 PjTf ≍f h and the Arzela - Ascoli Theorem, there is a uniformly

convergent subsequence of n−1∑n−1j=0 P

jTf . The same argument on the other

elements of H guarantee the existence of a subsequence of n−1∑nj=0P

jTf al-

most everywhere uniformly convergent, say, to F . The positive function Fwould satisfy F = PTF , so F = h almost everywhere. We can even take Fto be continuous when restricted to the interior of each Hi. With furthercomputations, we obtain the following proposition.

Proposition 3.2.14. There exists some f ∈ L(H) such that f = h a.e..

The rest of the proof of Theorem 3.2.12 is getting a lower bound of PTϕfor functions ϕ ∈ L(H). Once again, the argument focuses on full cylindersvia Lemma 3.1.2.

3.3 General frameworks

Even though the ergodic theory of Hurwitz Continued Fractions is an activesubject, most of the related research has been done in more general frame-works. First, we discuss the spaces where F. Schweiger got his Kuzmin-typetheorem. Afterwards, we describe the very recent work of A. Lukyanenkoand J. Vandehey. We already saw that the analogue of Serret’s Theoremon equivalent fractions does not hold for HCF. However, a consequence ofLukyanenko and Vandehey’s work is that it does hold almost everywhere.

55

3.3 General frameworks Ergodic Theory of Hurwitz Continued Fractions

3.3.1 Fibred Systems

A pair (B, τ) where B is a non-empty set and τ ∶ B → B is a fibred system ifthere is an at most countable partition of B, Bii∈I such that the restrictionsτ ∣Bi ∶ Bi → B are injective. We call I the set of digits. For any i1 ∈ I writeB1(i1) = Bi1 and define for any n ∈ N≥2 and any (i1, . . . , in) ∈ In

Bn(i1, . . . , in) ∶= B1(i1) ∩ τ−1[Bn−1(i2, . . . , in−1)].A cylinder of depth n is a set of the form Bn(i) for some i ∈ In. A finite se-quence (i1, . . . , in) ∈ In is valid if Bn(i1, . . . , in) ≠ ∅. A cylinder Bn(i1, . . . , in)is full if τn[Bn(i1, . . . , in)] = B. For any valid i = (i1, . . . , in) ∈ In define thefunction

V (i) ∶ τn[Bn(i)]→ B, z ↦ V (i; z)to be the inverse of τn restricted to Bn(i).

Let (B, τ) be a fibred system such that B ⊆ Rn is compact, connectedand mn(B) = 1 (mn is the n-dimensional Lebesgue measure); each Bi ismeasurable and connected, and τ is non-singular (with respect to mn). Wewill call a finite sequence i ∈ Im regular whenever mn (Bm(i)) > 0. Bythe Radon-Nikodym Theorem and the non-singularity of τ , for every regulari ∈ Im we can define ω(i) ∶ Bn(i)→ R by the condition

∀E ∈B(B) ∫Eω(i;x)dmn(x) = ∫

τ−m[E]∩Bm(i) 1dm.

A fibred system (B, τ) is a Schweiger Fibred System if B ⊆ Rn iscompact, connected and mn(B) = 1; each Bi is measurable and connected, τis non-singular (with respect to mn), and conditions (S1) to (S7) below hold.

S1. The sequence σ(n) ∶= supi∈In ∣Bn(i)∣ satisfies σ(n)→ 0 as n→∞.

S2. (Finite range condition) There is a finite measurable collection U =U1, . . . , UN such that for every n ∈ N∀i ∈ In i is regular Ô⇒ ∃j ∈ [1..N] τn[Bn(i)] = Uj.

The previous equality is understood up to a null set. We call F thepartition of B mod 0 induced by the intersections of the members of U .

S3. (Renyi Condition) There exists C > 0 such that for any regular i ∈ Iness sup

zω(i; z) ≤ C ess inf

wω(i;w).

56

Ergodic Theory of Hurwitz Continued Fractions 3.3 General frameworks

S4. Every U ∈ U contains a full cylinder.

S5. Every ω(i) is equal to a Lipschitz continuous function almost everywhere.

S6. For every regular i ∈ In, V (i) is Lipschitz continuous.

S7. Define for every m ∈ NLm ∶= a ∈ Im ∶ ∀F ∈ F Bm(a) /⊆ F, γ(m) ∶= ∑

i∈Lmmn (Bm(i)) .

Then, γ(m)→ 0 as m→∞.

Conditions S1, S2, S3, S4 imply the existence of a τ -invariant measure µequivalent to mn. The referred four conditions also guarantee the existenceof a function h which is Lipschitz continuous on each F ∈ F and such thatdµ = hdmn.

The seven conditions S1 − S7 allowed F. Schweiger to show his Kuzmin-type theorem (Theorem 3.3.1 below). Let L(F) be the space of functionsf ∶ B → R>0 such that

∃m,M > 0 m < infx∈B f(x) ≤ sup

x∈B f(x) <M,

∀F ∈ F f ∣F is Lipschitz continuous.

Theorem 3.3.1 (F. Schweiger, 2001). Let (B, τ) be a Schweiger Fibred Sys-tem with τ -ergodic measure µ and call h its density. Then, there is some0 < α < 1 for which

∀f ∈ L(H) ∀n ∈ N ∥P nτ f − h∫

Bfdmn∥∞ ≪f α

√n + σ([√n]) + γ([√n]).

The norm ∥ ⋅ ∥∞ refers to L∞(B,mn) and Pτ ∶ L1(B,mn)→ L1(B,mn) is thetransfer operator of τ with respect to mn.

As an application of the previous theorem, H. Nakada and H. Natsuiproved a mixing property of Schweiger Fibred Systems.

Theorem 3.3.2 (H. Nakada, R. Natsui, 2003). Let (B, τ) be a SchweigerFibred System and µ its ergodic measure. Define for every t ∈ NΨ(t) ∶= sup∣µ(Bs(i) ∩ τ−(s+t)[E] − µBs(i)µE∣

µBs(i)µE ∶ s ∈ N, mBs(i) > 0, µE > 0 .Then, Ψ(t)→ 0 as t→∞ and supt Ψ(t) < +∞.

Any fibred system satisfying the conclusion of Theorem 3.3.2 is calledcontinued-fraction mixing. Some statistical properties, such as Laws ofLarge Numbers for stationary processes arising from Schweiger Fibred Sys-tems, were also studied by H. Nakada and R. Natsui in [NaNa03].

57

3.3 General frameworks Ergodic Theory of Hurwitz Continued Fractions

3.3.2 Iwasawa Continued Fractions

Recently, A. Lukyanenko and J. Vandehey obtained the ergodicity of theGauss map for a large family of continued fractions algorithms. Their workinclude Hurwitz Continued Fractions and Julius’ Hurwitz Continued Frac-tions (discussed in the last chapter). We shall only state their results. Theproofs can be found in [LuVa18].

First, let k be either the real, the complex, or the quaternion numbersand n ∈ N. The Iwasawa inversion space X = Xn

k is the group given by theset kn × I(k) with the operation

(z, t) ∗ (w, s) = (z +w, t + s + 2I(⟨z,w⟩),with ⟨z,w⟩ = ∑j zjwj. We turn X into a metric space with the gauge function∣ ⋅ ∣ and the Cygan metric d defined by

∀(z, t), (w, s) ∈ X ∣(z, t)∣ = ∥∥z∥2 + t∥2, d((z, t), (w, s))) = ∣(−z,−t)∗(w, s)∣.

A conformal mapping ι ∶ X→ X satisfying

∣ιh∣ = 1∣h∣ , d(ιh, ιh′) = d(h,h′)∣h∣∣h′∣is an inversion. The notions of lattice, fundamental domain, and nearest-integer mapping can be extended to metric spaces with a measure invariantunder its isometries, in particular to X. An Iwasawa Continued FractionAlgorithm is defined by the next objects

i. An associative real algebra k and n ∈ N.

ii. The Iwasawa inversion space X = Xnk .

iii. An inversion.

iv. A discrete subgroup Z of the isometries of X, the associated fundamentaldomain K ⊆ X, and a nearest integer map ⌊⋅⌋ ∶ X → Z. The associatedGauss map T ∶K →K is defined by T (0) = 0 and T (x) = ⌊ι(x)⌋−1 ∗ ι(x).

As in the usual way, we can consider a continued fraction expansion. We callthe algorithm convergent if for almost every x ∈ X the associated continuedfraction converges to x.

An Iwasawa Continued Fraction is proper if the distance from K to theunit sphere is positive. Each Iwasawa Continued Fraction has associateda modular group M and whenever it is discrete, the Iwasawa ContinuedFraction is discrete. Moreover, the continued fraciton is complete if thestabilizer (over M) of a certain element is Z.

58

Ergodic Theory of Hurwitz Continued Fractions 3.4 Notes and Comments

Theorem 3.3.3 (A. Lukyanenko, J. Vandehey, 2018). If an Iwasawa Con-tinued Fraction is discrete and proper, it is convergent. If it is complete, thenit is ergodic.

The almost everywhere version Serret’s Theorem is the following result.

Theorem 3.3.4 (A. Lukyanenko, J. Vandehey, 2018). Let X be an Iwasawainversion space and consider a complete, discrete, and proper Iwasawa Con-tinued Fraction algorithm. For almost every pair x, y ∈ X the tails continuedfraction expansions coincide if and only if they belong to the same M-orbit.

In particular, as pointed out in [LuVa18], we have the following result forHCF.

Corollary 3.3.5. For almost every z = [0;a1, a2, . . .],w = [0; b1, b2, . . .] ∈ Fthere exists some M,N ∈ N such that aN+j = bM+j for all j ∈ N if and only ifAz = w for some A ∈ SL(2,Z[i]) acting via Mobius transformations.


1. The meaning we have established for regular, valid, and admissible dif-fers from that in the fibred systems literature. The standard terminologydoes not include the word regular, instead, only admissible is used. Whileavoiding the notion of regularity and irregularity works perfectly fine froma (Lebesgue) measure theoretic perspective, it must be taken into consid-eration when studying Hausdorff dimensions.

2. The ergodic theory of HCF is an active research field. For example, re-cently J. Vandehey and R. Hiary ([HiVa18]) gave some conjectures on hbased on their numerical experiments.

3. We have not found (3.2.6) in the available literature. However, we believethat it is very unlikely that this has not been shown before in more generalcontexts.

4. In his work on multi-dimensional continued fractions, F. Schweiger ob-tained a result analogous to Theorem 3.2.7. Nevertheless, as he pointedout, Hurwitz Continued Fractions are not part of this framework.

The approach on Theorem 3.2.7 is the one adopted by S. Tanaka whilestudying J. Hurwitz continued fractions (see Chapter 5).

5. S. G. Dani and A. Nogueira used the term generic instead of normalfor naming sequences having all regular prefixes as sub-blocks. We choseHCF-normal in analogy to regular continued fractions.

59

3.4 Notes and Comments Ergodic Theory of Hurwitz Continued Fractions

60

Chapter 4

Hausdorff Dimension Theory

In this chapter, we estimate the Hausdorff dimension of sets given by HCFrestrictions. First, we shift our attention to the real numbers. After recall-ing the definition of badly approximable real numbers, BadR, we discusswhy is it a null set. Motivated by our discussion, we introduce the no-tion of Hausdorff dimension. Afterwards, we discuss Jarnık’s argument forshowing dimH BadR = 1. Later, we review some generalizations and exten-sions of Jarnık’s work. In the following section, we fully focus again on thecomplex plane. First, we use a result of W.J. Leveque to establish thatmBadC = 0 and then we use the Kristensen-Thorne-Velani Theorem to showthat dimH BadC = 2. We later discuss a full HCF analogue of a theorem byI.J. Good. We finish the chapter by computing the dimension of subsets ofBadC where the laws of succession can somehow be overlooked.

Notation. During Section 1 and Section 2, we consider only regular con-tinued fractions. In Section 3, the continued fractions used are Hurwitz.Although we use the same notation for both types of continued fractions,there will be no room for ambiguity. Recall that for any j, k ∈ Z with j < kwe write [j..k] ∶= n ∈ Z ∶ j ≤ n ≤ k.When ≪ or ≍ appear without any additional parameter, the implied constantsare absolute.

Main references. For general theory of Hausdorff Dimension, see [Mat],[BiPe17]. For the theory of real continued fractions, Jarnık’s Theorem andits generalizations, see [Kh], [Ja28], [Goo41], [KTV06], [KlWe10]. For therelated theory for HCF: [Gon18-02]. For Schmidt Games, see [Sch66] and[BHNS18].

61

4.1 Hausdorff Dimension. Definitions and Elementary PropertiesHausdorff Dimension Theory

4.1 Hausdorff Dimension. Definitions and El-

ementary Properties

Regular continued fractions provide for any irrational number α ∈ R infinitelymany pairs (p, q) ∈ Z ×N satisfying

∣α − pq∣ < 1

q2.

The number α is badly approximable if the previous inequality is best pos-sible; that is, α ∈ R is badly approximable if

∃C > 0 ∀(p, q) ∈ Z ×N ∣α − pq∣ > c

q2.

BadR denotes the set of badly approximable real numbers. As we mentionedbefore, a classical theorem characterizes BadR in terms of their continuedfraction expansion:

BadR ∶= α = [a0;a1, a2, a3, . . .] ∈ R ∖Q ∶ lim supan < +∞(see Theorem 23, p. 36, in [Kh]). Thus, we can express BadR as a countableunion of increasing sets: BadR = ⋃M FM , where

∀M ∈ N FM = [0;a1, a2, a3, . . .] ∶ ∀n ∈ N an ≤M .In 1909, E. Borel showed that m1BadR = 0 (see [Bo09]). Today we know sev-eral arguments showing Borel’s equality. We briefly sketch one particularlyillustrative to our ends.

i. It suffices to show m(FM) = 0 for any M ∈ N. Fix M ∈ N. For any a ∈ Nn,let Cn(a) be the closure of the set of irrationals in [0,1] whose regularcontinued fraction starts with a. We call these sets cylinders of depthn. Define the collection of compact sets

∀n ∈ N F nM ∶= ⋃

a∈[1..M]n Cn(a).ii. Evidently, an upper bound on m(Cn(a)), a ∈ [1..M]n, entails an upper

estimate of m(F nM). Using the estimate

∀a ∈ Nn ∀j ∈ N m (Cn+1(aj))m (Cn(a)) ≍ 1

j2, (4.1)

62

Hausdorff Dimension Theory4.1 Hausdorff Dimension. Definitions and Elementary Properties

we may bound m(F n+1M ) in terms of m(F n

M). Indeed, for any a ∈ [1..M]nm (Cn(a)) = M∑

j=1

m (Cn+1(a, j)) + ∞∑j=M+1

m (Cn+1(a, j))≥ M∑j=1

m (Cn+1(a, j)) + 1

3( ∞∑j=M+1

1

j2)m (Cna) .

Adding along a ∈ [1..M]n, we deduce the existence of some 0 < τ =τ(M) < 1 such that that m1(F n+1

M ) ≤ τm1(F nM) for every n. As a conse-

quence, m1F n+1M → 0 as n→∞ and mFM = 0.

To sum up, we concluded mFM = 0 thanks to a uniform relation amongm (Cn(a)) and m (Cn+1(aj)) ∶ 1 ≤ j ≤M.

4.1.1 Definitions

Even though the Lebesgue Measure has undoubtedly proven to be extremelyuseful, it has some drawbacks. For instance, the structure of null sets iscompletely overlooked. In particlar, while F1 (as above) contains only onepoint, Bad is— although meager— an uncountable dense subset of [0,1].

A slight modification of Caratheodory’s construction of Lebesgue’s mea-sure leads to the s-dimensional Hausdorff measures and to the Hausdorffdimension. These objects help to refine dramatically the notion of null sets.

Recall that for a metric space (X,d) and A ⊆ X, we denote by ∣A∣ thediameter of A: ∣A∣ = supd(a, b) ∶ a, b ∈X if A ≠ ∅ and ∣∅∣ ∶= 0.

The s-Hausdorff measures are constructed with countable coverings formedby small sets. To be more precise, for any δ > 0 a δ-cover of A is a countablefamily of subsets of X, (Aj)j≥1, such that

A ⊆ ∞⋃j=1

Aj, ∀j ∈ N ∣Aj ∣ ≤ δ.We define Hausdorff measures and dimension as in [Mat], p.54.

Definition 4.1.1. Let B be a family of subsets of a separable and completemetric space (X,d), and ζ ∶ B → R≥0. Suppose that the following conditionshold:

i. for any δ > 0 there is a δ-covering of X contained in B,

ii. for every δ > 0 there is some B ∈ B such that max∣B∣, ζ(B) < δ.63

4.1 Hausdorff Dimension. Definitions and Elementary PropertiesHausdorff Dimension Theory

For any δ > 0, define ψδ ∶ 2X → [0,+∞] by

∀A ⊆X ψδ(A) ∶= inf ∞∑j=1

ζ(Aj) ∶ Aj∞j=1 ⊆ B is a δ − covering of A .The outer measure ψ = ψ(B, ζ) is given by

∀A ⊆X ψ(A) ∶= limδ→0

ψδ(A).For s ≥ 0, the s-dimensional Hausdorff measure Hs is the outer measureobtained by setting B = 2X , and ζ(A) = ∣A∣s for any A ∈ B.

We adopt V. Jarnık’s convenient notation Λs. If A is a countable familyof subsets of X and s ≥ 0, we write

Λs(A) ∶= ∑A∈A ∣A∣s.

For example, for any A ⊆X we have

∀δ > 0 Hsδ(A) = infΛs(A) ∶ A is a δ − cover of A.Fix, for a moment, a set A ⊆ X and take 0 ≤ s < t, 0 < δ < 1. Let A = (Aj)j≥1

be a countable δ-cover of A, then

Λt(A) =∑j≥1

∣Aj ∣t =∑j≥1

∣Aj ∣s∣Aj ∣t−s ≤ δt−s∑j≥1

∣Aj ∣s = δt−sΛs(A).By taking infima, Htδ(A) ≤ δt−sHsδ(A), then Hs(A) < +∞ implies Ht(A) = 0;equivalently, Hs(A) = +∞ whenever Ht(A) > 0. Hence, s ↦ Hs(A) takes atmost three values. Moreover, s ↦ Hs(A) is first constant and equal to +∞,a jump happens, and afterwards it is equal to 0. We are interested in thepoint where the jump occurs, called the Hausdorff dimension of A.

For our goals, it suffices to heed subsets of Rn.

Definition 4.1.2. Let A ⊆ Rn. The Hausdorff dimension of A is

dimH A = infs ∶ HsA = 0.Proposition 4.1.1. Let A,B ⊆ Rn.

i. 0 ≤ dimH A ≤ n.

ii. A ⊆ B Ô⇒ dimH A ≤ dimH B

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Hausdorff Dimension Theory 4.2 Jarnık’s Theorem

iii. If (An)∞n=1 is a sequence of subsets of X, then

dimH ( ∞⋃n=1

An) = supdimH An ∶ n ∈ Niv. Let j, n be integers with 1 ≤ j ≤ n. There is a constant k(j) such thatHj = k(j)mj, where mj is the j-th dimensional Lebesgue measure.

The previous proposition is well known and can be found, for instance,in the fourth chapter of [Mat].

4.2 Jarnık’s Theorem

In 1909, E. Borel’s Theorem showed that BadR is small; in 1929, V. Jarnıkshowed that not that much: dimH BadR = 1. As with mBadR = 0, there havebeen several improvements on the techniques required for showing Jarnık’sresult. Most notably, Wolfgang Schmidt introduced in 1965 a techniqueknown today as Schmidt games (discussed below). W. Schmidt obtained asufficient condition (positive winning dimension) for a set to have the samedimension as the ambient space. Rather than the more modern machinery,we study Jarnık’s original argument.

Theorem 4.2.1 (V. Jarnık, 1929). For any M ∈ N at least 9

1 − 4

M log 2≤ dimFM ≤ 1 − 1

8M logM.

4.2.1 Jarnık’s Strategy

After nearly a century from its publication, Jarnık’s approach still standsas a paradigm for Hausdorff dimension estimation. Let M ≥ 9 be a naturalnumber. We divide the argument into two major parts. First, we replace ar-bitrary coverings by coverings formed by sets Cn(a). Afterwards, by relating∣Cn(a)∣ with ∣Cn+1(aj)∣ ∶ 1 ≤ j ≤M in a uniform way (on n ∈ N and a ∈ Nn),we obtain upper and lower bounds for dimH FM .

A natural measure

Let M > 1 be a natural number. Define the family of compact sets Kna ∶ n ∈

N,a ∈ [1..M]n via

∀n ∈ N ∀a ∈ [1..M]n Kna ∶= M⋃

j=1

Cn+1(aj).65

4.2 Jarnık’s Theorem Hausdorff Dimension Theory

From the elementary continued fraction theory,

∣Cn(a)∣ ≍M d(Kn+1aj ,Kn+1

ak ). (4.2)

holds for any n ∈ N, a ∈ [1..M]n and j, k ∈ [1..M], j ≠ k.Take 0 < δ < min∣C1(a)∣ ∶ 1 ≤ a ≤M and let G be a finite open δ-covering

of the compact set FM . To each G ∈ G associate the cylinder C = C(G) ofmaximum depth containing G. The maximality of C(G) and (4.2) imply∣G∣ ≍s ∣C(G)∣. Call C(G) the cover obtained by replacing each G ∈ G by thecorresponding C(G). Then, adding over G, Λs(G) ≍s Λs(C(G)). Taking theinfimum over the family of open finite coverings and letting δ tend to 0, weobtain the left inequality in Lemma 4.2.2 below. The right inequality followsfrom the definition of the outer measures.

Lemma 4.2.2. Take s ≥ 0 and define

B = Cn(a) ∶ n ∈ N,a ∈ Nn, ∀A ∈ B ζs(A) = ∣A∣s.Let Hs = ψ(B, ζs) be the resulting outer measure (as in Definition 4.1.1).Then, for some B(s) > 0

B(s)Hs(FM) ≤ Hs(FM) ≤ Hs(FM), (4.3)

and dimH FM = infs ≥ 0 ∶ Hs(FM) = 0.

A lower bound

Lemma 4.2.3 (First Jarnık Lemma). Let 0 < s < 1 and M ≥ 2 be given. Iffor every a ∈ [1..M]n−1 we have

∣Cn−1(a)∣s ≤ M∑k=1

∣Cn(ak)∣s, (4.4)

then dimH FM ≥ s.Proof. Given δ > 0, let G be a finite δ-covering of FM formed by pairwisedisjoint cylinders. Let n be the maximal integer N such that G contains acylinder of depth N . Take a ∈ [1..M]n−1 and j ∈ [1..M] such that Cn(aj) ∈ G.We must have Cn(ak) ∶ k ∈ [1..M] ⊆ G, since G covers FM and its elementsare pairwise disjoint. Let G′ be the covering obtained by replacing in G thesets Cn(ak) ∶ k ∈ [1..M] by Cn−1(a); hence, (4.4) gives Λs(G′) ≤ Λs(G).After iterating the argument we eventually arrive at 0 < Λs([0,1]) ≤ Λs(G),so Hs(FM) > 0 and, by Lemma 4.2.2, dimH FM ≥ s.

66


An upper bound

Lemma 4.2.4 (Second Jarnık Lemma). Let 0 < s < 1 and M ≥ 2 be given.If for every n ∈ N and all a ∈ [1..M]n−1 we have

M∑k=1

∣Cn(ak)∣s ≤ ∣Cn−1(a)∣s, (4.5)

then dimH FM ≤ s.Proof. Take δ > 0. Since ∣Cn(a)∣ ≤ 2−n2 , the family Gn = Cn(b) ∶ b ∈ [1..M]nis a δ-covering of FM for any large n. Fix such an n. Let G′n be the coveringof FM obtained by replacing Cn(aj) ∶ 1 ≤ j ≤ M by Cn−1(a) for each a ∈[1..M]n−1. By (4.5), Λs(G) ≤ Λs(G′) and repeating the process we eventuallyobtain Λs(G) ≤ Λs([0,1]) = 1. Hence, Hs(FM) < +∞ and, by Lemma 4.2.2,dimH FM ≤ s.Proof of Theorem 4.2.1

Proof of Theorem 4.2.1. Let M,n ∈ N satisfy n ≥ 2, M ≥ 9. Take a ∈ [1..M]nand let (qn)n≥0 be the associated sequence. In view of ∣Cn−1(a)∣ = (qn−1(qn−1+qn−2))−1, there is some τ , 2−1 < τ < 2, such that

M∑k=1

∣Cn(ak)∣ = ∣Cn−1(a)∣ (1 − τ

M)

Writing ∣Cn(ak)∣s = ∣Cn(ak)∣∣Cn(ak)∣s−1 and calculating, we get

M∑k=1

∣Cn(ak)∣s ≥ 21−s (1 − 2

M) ∣Cn−1(a)∣s.

The hypothesis of the First Jarnık Lemma (Lemma 4.2.3) are met if thecoefficient of ∣Cn−1(a)∣s exceeds 1, which occurs if and only if (1 − s) log 2 ≥− log(1 − 2/M). With the aid of the Taylor expansion of the logarithm, wesee that s = 1 − (M log 2)−1 works.

The upper bound is treated in a similar fashion with Lemma 4.2.4 insteadof Lemma 4.2.3.

4.2.2 Extensions, abstractions, and generalizations

Except for M = 1, the sets FM are are homeomorphic to a countable prod-uct of finite topological spaces. Moreover, cylinders provide the appropriatecompact sets to obtain each FM through a process resembling the construc-tion of the middle-third Cantor set. In both, Borel’s Theorem and Jarnık’s

67


Theorem, this feature was heavily exploited by relating two consecutive it-erations (cfr. (4.1) and (4.4),(4.5)). This is the key for the extensions weregard.

Generalized Jarnık Lemmas

The first extension of Jarnık’s Lemmas we will consider take place in completemetric spaces. We will estimate the Hausdorff dimension of sets obtained byslightly modifying the notion of strongly tree-like sets as defined in [KlWe10].

Definition 4.2.1. Let (X,d) be a complete metric space. A family of com-pact sets A is diametrically strongly tree-like if A = ⋃∞

n=0An where eachAk is finite, #A0 = 1, and

i. ∀A ∈ A ∣A∣ > 0,

ii. ∀n ∈ N ∀A,B ∈ An (A = B) ∨ (A ∩B = ∅),

iii. ∀n ∈ N ∀B ∈ An ∃A ∈ An−1 B ⊆ A,

iv. ∀n ∈ N ∀A ∈ An−1 ∃B ∈ An B ⊆ A,

v. dn(A) ∶= max∣A∣ ∶ A ∈ An→ 0 as n→∞.

The n-th stage diameter is dn(A). The limit set of A, A∞, is

A∞ ∶= ∞⋂n=0

⋃A∈AnA.

The First Generalized Jarnık Lemma (Lemma 4.2.5 below) gives a lowerbound on the limit set of a family of diametrically strong-tree like compactsets. In the statement we have merged the conditions for enabling the gener-alization of Lemma 4.2.2 and 4.2.3. We also consider a restriction weaker than(4.4). An upper bound is given by the Second Generalized Jarnık Lemma(Lemma 4.2.6 below). On the basis of our discussion of Jarnık’s argument,the proofs Lemmas 4.2.5 and Lemma 4.2.6 are now a simple exercise (for thedetails, see [Gon18-02]).

Lemma 4.2.5 (First Generalized Jarnık Lemma). Let A be a diametricallystrongly tree-like family of compact sets. Suppose that for some κ > 1

∀n ∈ N0 dn(A) ≤ 1

κn. (4.6)

68


Assume the existence of a sequence (Bn)n≥1 with 0 < Bn ≤ 1 for all n,

lim supn→∞

log log(B−1n )

logn< 1, (4.7)

and such that

∀n ∈ N ∀A ∈ An ∀Y,Z ∈ An+1 (Y ∪Z ⊆ A & Y ≠ Z Ô⇒ d(Y,Z) ≥ Bn∣A∣).(4.8)

If for s > 0 there exists some c > 0 such that

∀X ∈ 2A (#X < +∞ & A∞ ⊆ ⋃A∈XA Ô⇒ ΛsX > c) , (4.9)

then dimH A∞ ≥ s.Kepping the lemma’s notation, we can state an analogue of (4.4):

∀n ∈ N ∀A ∈ An ∑A′⊆A

A′∈An+1∣A′∣s ≥ ∣A∣s. (4.10)

An iterative replacement argument, like the ones in Jarnık’s lemmas, tells usthat (4.10) implies (4.9).

Lemma 4.2.6 (Second Generalized Jarnık Lemma). Let A be a family ofcompact sets satisfying conditions i., iii., iv., v. of the definition of stronglytree-like structure and such that the each Ak, k ≥ 1, is at most countable. Lets > 0.

If for every k ∈ N and every A ∈ Ak∑B

∣B∣s ≤ ∣A∣s,where the sum runs along the sets B ∈ Ak+1 satisfying B ⊆ A, then dimH A∞ ≤s.

The Kristensen-Thorn-Velani Theorem

Jarnık’s Theorem was established in a quite abstract context by Simon Kris-tensen, Rebecca Thorn, and Sanju Velani ([KTV06]). In general, computingthe Hausdorff Dimension of a given set can be a daunting task. While upperestimates can be obtained by choosing the right coverings, lower bounds canbe much more difficult to get. A useful strategy— followed also by the afore-mentioned authors— for estimating the Hausdorff dimension of a specific setis constructing an appropriate probability measure.

69


Theorem 4.2.7 (Mass Distribution Principle). Let (X,d) be a completemetric space and F ⊆X. If F supports a measure µ such that for some s ≥ 0,and c, ρ > 0

µB(x, r) ≤ crs,then dimH F ≥ s.Proof. Take 0 < 2r < ρ and let (En)∞n=1 be an r-cover of F . For each En takea ball Bn = B(xn,diamEn) containing En, then

µF ≤ µ( ∞⋃n=1

Bn) ≤ ∞∑n=1

µBn ≤ c ∞∑n=1

(diamEn)s Ô⇒ ∞∑n=1

(diamEn)s ≥ µFc

> 0.

Hence, the infimum over all r-coverings is positive and dimH F ≥ s. .

Let us establish the framework for the Kristensen-Thorne-Velani Theo-rem. Let (X,d) be a complete metric space, Ω ⊆ X a compact subset, andm a non-atomic Borel measure supported in Ω. Consider a countable familyof subsets of X, called resonant sets, R ∶= Rα ∶ α ∈ J. Let β ∶ J → R>0

satisfy ∀M > 0 #β−1[(0,M]] < +∞,and ρ ∶ R>0 → R>0 be such that

limr→∞ρ(r) = 0, ∃R > 0 ∀r, s ∈ R>R (r < s Ô⇒ ρ(s) < ρ(r)).

The badly approximable elements, Bad∗(R, β, ρ), are the points which stayaway (in terms of ρ) from resonant sets:

Bad∗(R, β, ρ) ∶= x ∈ Ω ∶ ∃c > 0 ∀α ∈ J cρ(βα) ≤ d(x,Rα).In our scenario, the measure m and the function ρ satisfy the following con-ditions:

(A) There are some r0, δ > 0 such that for some absolute constants

∀c ∈ Ω ∀r ∈ (0, r0] m(B(c, r)) ≍ rδ.(B) There are functions λl, λu ∶ R>0 → R>0 such that λl(k) → ∞ as k → ∞

and some K > 0 with

∀k ∈ R≥K ∀n ∈ N λl(k) ≤ ρ(kn)ρ(kn+1) ≤ λu(k).

70

Hausdorff Dimension Theory 4.3 A word on Schmidt Games

If k > 1 is fixed, define

∀n ∈ N J(n) = β−1[[kn−1, kn)]Whenever B ⊆X, c(B) ∈X will denote a center of B; i.e., for some r > 0 wehave B = B(c(B), r).Theorem 4.2.8 (S. Kristensen, R. Thorn, S. Velani (2006)). Let (X,d,Ω,m,β, ρ)be as above. Suppose that m satisfies Condition (A) and that ρ satisfies Con-dition (B). Assume there is some k0 > 1 such that for every k ≥ k0 there existsθ = θ(k) > 0 satisfying the following:

For every n ∈ N and all c ∈ X, Bn ∶= B(c;ρ(kn)), there is a collectionC(Bn) of pairwise disjoint balls of radius 2θρ(kn+1) such that

⋃B∈C(Bn)B ⊆ B(c, θρ(kn)),

and for some numbers 0 < κ1 < κ2 independent of k and n we have

κ1 ( ρ(kn)ρ(kn+1))

δ ≤ #C(Bn) (4.11)

and

#B ∈ C(Bn) ∶ minα∈J(n+1)d(c(B),Rα) ≤ 2θρ(kn+1) ≤ κ2 ( ρ(kn)

ρ(kn+1))δ

. (4.12)

Additionally, suppose that for the δ of Condition (A) we have dimH ⋃α∈J Rα <δ, then

dimH Bad∗(R, β, ρ) = δ.It can be checked (see Example 1 in [KTV06]), that Theorem 4.2.8 implies

dimH BadR = 1.

4.3 A word on Schmidt Games

Wolfgang M. Schmidt introduced in [Sch66] a useful game to estimate theHausdorff dimension of some sets. We comment them only in their simplestform. To start, fix a set S ⊆ Rn. The game is played between two players, Aand B, and exactly one of them will win. Assign to A and B, respectively, anumber α,β in (0,1). The game starts with B choosing a compact ball B1

of radius ρ1 > 0. Then, A chooses a compact ball B2 ⊆ B1 of radius ρ2 ∶= αρ1.Afterwards, B chooses a compact ball B3 ⊆ B2 of radius ρ3 = βρ2 and continue

71

4.3 A word on Schmidt Games Hausdorff Dimension Theory

infinitely. The sequence of radii, (ρn)n≥1, tends to 0, hence intersection ⋂nBn

contains only one point s. A wins if s ∈ S; otherwise, B wins.A set S ⊆ Rn is (α,β)-winning if no matter how B plays, A can find

a way to win. For a given 0 < α < 1, we say that S is α-winning if it is(α,β)-winning for every 0 < β < 1. It can be shown (Lemma 11 in [Sch66])that if S is α-winning, then it is α′-winning for every 0 < α′ < α. It is thusnatural to define the winning dimension of S, win dimS, as

win dimS ∶= supα > 0 ∶ S is α-winningwhen the referred set is non-empty and win dimS ∶= 0 otherwise. We call aset with positive winning dimension winning.

Winning sets possess desirable closure properties.

Proposition 4.3.1. Let S, (Sj)j≥1 be subsets of Rn and α > 0.

i. If win dimS > 0, then dimH S = n.

ii. If every Sn is α-winning, then ⋂n Sn is α-winning.

iii. If S is winning, then it is imcompressible; that is, for every non-emptyopen set U ⊆ Rn and every sequence of uniformly bi-Lipschitz maps(fn)n≥1, fn ∶ U → Rn,

dimH (⋂n≥1

f−1n [S]) = n.

The first two statements are proven in [Sch66] as Corollary 2 and Theorem2, respectively. The third one is shown in [Da89] as Theorem 1.1.

W. M. Schmidt showed that win dimBadR = 12 . In the complex plane,

the analogous result was obtained by M. Dodson and S. Kristensen for BadC(Theorem 5.2. in [DoKr]). A further extension was obtained by R. Esdahl-Schou and S. Kristensen in [EK10]. For each

D ∈ 1,2,3,5,7,11,19,43,67,163define

BadD ∶= z ∈ C ∶ ∃K > 0 ∀(p, q) ∈ Z [√−D] ×Z [√−D]∗ ∣z − pq∣ ≥ K∣q∣2 .

Theorem 4.3.2 (R. Esdahl-Schou, S. Kristensen, 2009). Let K ⊆ C be acompact supporting a measure µ satisfying

∀z ∈K ∀r ∈ (0, r0) arδ ≤ µ(D(z, r)) ≤ brδ,72

Hausdorff Dimension Theory 4.4 Hausdorff Dimension for HCF Sets

where a, b, δ, r0 > 0 are constant. Then, any

∅ ≠ E ⊆ 1,2,3,5,7,11,19,43,67,163verifies

win dim(K ∩ ⋂D∈EBadD) > 0.

It trivially follows from Theorem 4.3.2 that

dimH BadC = dimH Bad1 = 2.

A recent survey on some variations of Schmidt games and their relation canbe found in [BHNS18].

4.4 Hausdorff Dimension for HCF Sets

Let us return, after our short digression, to the complex plane and HCF.Once again, [a0;a1, a2, . . .] will represent a HCF.

4.4.1 Measure and Dimension of BadC

W. J. Leveque developed a complex continued fraction algorithm in [Le52a]and [Le52b]. In the second paper, he could show an analogue of a famousresult by A. Khinchin (Theorem 1.1 in [Bu04], or Theorem 32 in [Kh]). Weneed some terminology for stating Leveque’s result. For any continuous andnon-increasing function Ψ ∶ R>0 → R>0, the set of Ψ-well approximablenumbers is

W (Ψ) = z ∈ F ∖Q(i) ∶ ∣z − pq∣ < Ψ(q) for infinitely many (p, q) ∈ Z[i] ×Z[i]∗ .

Although we can surely have defined W (Ψ) as a subset of C, for notationalconvenience we opted not to. It should be clear that there is absolutely noloss of generality by doing this.

Theorem 4.4.1 (W. J. Leveque (1952)). Let f ∶ R>0 → R>0 be a continuousfunction such that t↦ t2f(t) is non-increasing. Then,

mWC(Ψ) =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0, if ∑n≥1

n3Ψ(n)2 < +∞,1, if ∑

n≥1

n3Ψ(n)2 = +∞.73

4.4 Hausdorff Dimension for HCF Sets Hausdorff Dimension Theory

Proof. See Theorem 7, [Le52b].

The convergence part is an easy exercise on the Borel-Cantelli Lemma.The usual argument for the real version of Khinchin’s Theorem (Theorem1.10 in [Bu04]) relies on the existence of the Khinchin-Levy constant and theBorel-Bernstein Theorem. In [Le52a] and [Le52b], Leveque obtained similarpropositions that permitted him to conclude Theorem 4.4.1.

Let Ψ ∶ R≥0 → R>0 be a continuous function such that Ψ(x) = (x2√

logx)−1

for x ≥ 2. Since F ∖WC(Ψ) ⊇ BadC ∩ F, Theorem 4.4.1 yields

mBadC = 0.

Thus, asking for the Hausdorff dimension of BadC is not futile. We havealready obtained from Theorem 4.3.2 that dimH BadC = 2. However, we giveanother proof, already pointed out in [KTV06], based on Theorem 4.2.8.

Theorem 4.4.2. dimH BadC = 2.

Proof. Recall that ClA denotes the closure of a set A. Define

X ∶= Ω ∶= ClF, J = Z[i] ×Z[i]∗, ∀α = (p, q) ∈ J Rα ∶= pq ,

m = m, ∀(p, q) ∈ J β(p, q) = q, ∀k > 0 ρ(k) = k−2,

κ1 = 1

25, κ2 = 1

16, δ = 2, k0 = 6.

Consider the metric d∞ given by

∀z,w ∈ C d∞(z,w) = max∣R(z −w)∣, ∣I(z −w)∣and denote by d∞ its restriction to Ω =X.

We now verify Condition (A). Let B(z, r) denote the ball with center zand radius r with respect to the metric d∞. Take z ∈X, r ∈ (0, 1

2) and divideBd∞(z, r) into four equal squares in the natural way. Note that Bd∞(z, r)contains at least one of this quarters, so

∀z ∈ Ω ∀r ∈ (0,0.5) mB(z, r)rδ

= mB(z, r)r2

≍ 1,

with the implied constants absolute (in fact, 4−1 and 4 work). Condition (B)is evident by taking λl(k) = λu(k) = k2 for any k > k0.

We now check the additional hypothesis of Theorem 4.2.8. First, notethat for k > k0 and any n ∈ N we clearly have

( ρ(kn)ρ(kn+1))

δ = k4.

74

Hausdorff Dimension Theory 4.4 Hausdorff Dimension for HCF Sets

Let α = (p, q) and α′ = (p′, q′) belong to J(n + 1) with Rα ≠ Rα′ , then

d∞(Rα,Rα′) ≥ 1√2∣pq− p′q′ ∣ ≥ 1√

2

1∣qq′∣ > 1√2

1

k2n+2> 1

4k2n+2. (4.13)

For any k > k0 define θ(k) = 14k

−2. Thus, for any c ∈ Ω the ball Bn =B(c, θρ(kn)) contains at most one point of J(n + 1). Suppose we had

Bn ⊇ c + is + t ∈ C ∶ 0 ≤ s, t < k−2n−2

4 ,

(if this did not hold, we just take the appropriate square by considering ±s,±t). By direct computations, we see that the set to the right contains at

least k4

16 = κ1 ( φ(kn)φ(kn+1))δ squares with sides of length 2θρ(kn+1). Also, in view

of (4.13), the term at the left in (4.12) is at most 1 ≤ κ2k4.Finally, since dim (⋃αRα) = 0 < δ, Theorem 4.2.8 guarantees

dimH BadC = dimBad∗(R, β, ρ) = δ = 2.

4.4.2 Good’s Theorem for HCF

The equivalence between the boundedness of the HCF expansion and belong-ing to BadC allows us to rewrite Theorem 4.4.2 as

dimH z = [a0;a1, a2, a3, . . .] ∈ C ∶ lim supn→∞ ∣an∣ < +∞ = 2.

In 1941, I. J. Good obtained that the Hausdorff dimension of the set ofirrationals whose regular continued fraction tends to infinity is 1

2 . We canmodify Good’s argument– which is itself a modification of Jarnık’s– to deducea full analogue of his theorem for HCF.

Theorem 4.4.3.

dimH z = [a0;a1, a2, a3, . . .] ∈ C ∶ limn→∞ ∣an∣ = +∞ = 1.

We merely sketch the proof, the details are in [Gon18-02]. As notedbefore, it is enough to work in F. Define the set

E = ζ = [0;a1, a2, . . .] ∈ F ∶ limn→∞ ∣an∣ = +∞.

In order to obtain an upper bound of dimH E, we consider

∀L ∈ N E′L ∶= ζ = [0;a1, a2, . . .] ∈ F ∶ lim inf

n→∞ ∣an∣ ≥ L,∀L,k ∈ N E′

L(k) ∶= ζ = [0;a1, a2, . . .] ∈ F ∶ ∀n ∈ N0 ∣an+k∣ ≥ L.75

4.5 Independent Blocks Hausdorff Dimension Theory

Lemma 4.4.4. limL→∞ dimH E′L(1) ≤ 1.

Lemma 4.4.5. Let f, g ∶ N → R>0 be functions such that for some fixed cwith 0 < c < 1 we have

√8 ≤ f(n) < (1 − c)g(n) and g(n)2 ≤ en for all n ∈ N.

Then, dimH Ef,g = 1.

For every L, (E′L(k))k≥1 is an increasing sequence converging to E′

L, so

limk→∞dimH E

′L(k) = dimH E

′L.

However, by the invariance of the Hausdorff dimension under bi-Lipschitzmaps, dimH E′

L(k) = dimH E′L(1) and, thus, dimH E′

L = dimH E′L(1) for all k.

Taking limits, dimH Ek(1)′ = dimH E′M . As a consequence, by E = ⋂M E′

M ,dimH E ≤ dimH E′

M for all M . Lemma 4.4.4, thus, gives

dimH E′ ≤ 1.

The lower bound, as usual, is trickier. The idea is to approximate E′ fromthe inside by sets where we can control the growth of ∣an∣. To be more precise,given f, g ∶ N→ R>0 with f ≤ g, define

Ef,g = ζ = [0;a1, a2, a3, . . .] ∶ ∀n ∈ N f(n) ≤ ∥an∥ ≤ g(n),where ∥z∥ = max∣Rz∣, ∣Iz∣ for any z ∈ C.

Clearly, f(n)→∞ as n→∞ implies Ef,g ⊆ E. Lemma 4.4.5 tells us thatunder certain restrictions, dimH Ef,g = 1, so

1 ≤ dimH E.

The Generalized Jarnık Lemmas lie in the heart of the proof of Lemmas 4.4.4and 4.4.5.

4.5 Independent Blocks

Throughout this work we have seen that similarities between regular andcomplex continued fractions abound. We have pointed out important differ-ences too. Most notably, we have contrasted the intricate structure of ΩHCF

R

against NN. The complications in ΩHCF disappear when we consider onlysequences (an)n≥1 satisfying minn ∣an∣ ≥ √

8. It is natural to investigate thesubsets of BadC where uniform lower bounds are imposed. In this direc-tion, the proof of Theorem 4.4.3 yields the next corollary (see [Gon18-02] fordetails).

76

Hausdorff Dimension Theory 4.5 Independent Blocks

Corollary 4.5.1. For any L ∈ N with L ≥ √8 define

EL ∶= z = [0;a1, a2, a3, . . .] ∈ F ∶ ∀n ∈ N L ≤ ∣an∣ ,then

limL→∞dimH (EL ∩BadC) = 1.

Working with blocks, rather than individual terms, we obtain larger sub-sets of BadC. For z ∈ C write M (z) ∶= min∣Rz∣, ∣Iz∣ and for any j0,M, l ∈ Ndefine

A(j0,M, l) ∶= (an)n≥1 ∈ ΩHCFR ∶ ∀n ∈ N ∣an∣ ≤M, j0 ≤ M (an(l+1)) ,F(j0,M, l) ∶= Φ[A(j0,M, l)].

We devote the rest of the chapter to show the next result. Note that, inthis case, we can naturally interpret the blocks of length l+1 as independentidentically distributed random variables.

Theorem 4.5.2. If j0 ∈ N satisfies j0 ≥ 3, then

supl,M∈NdimH F(j0,M, l) = 2.

4.5.1 Preliminary Lemmas

The mere definition of F(j0,M, l) makes clear how to define the appropriatediameter tree-like structure. Our first preliminary result, Lemma 4.5.3, isa particular case of the First Generalized Jarnık Lemma. The verificationof the hypothesis hides some delicate details persuading us to give thoroughexposition.

Lemma 4.5.3. Fix j0,m,M ∈ N with j0 ≥ 3. Let s > 0 be such that for r ∈ Nand any a ∈ A(j0,m,M)[1..r(m+1)] we have that

∑c

∣C(r+1)(m+1)(a;c)∣s ≥ ∣Cr(m+1)(a)∣s > 0, (4.14)

where c runs along A(j0,m,M)[1..m + 1]. Then, dimH F(j0,m,M) ≥ s.Recall that the chordal metric ρ is a metric on C ∶= C ∪ ∞, given by

∀w, z ∈ C ρ(z,w) = 2∣z −w∣√1 + ∣z∣2√1 + ∣w∣2 ;

if w =∞ and z ∈ Cρ(z,w) = 2√

1 + ∣z∣2 ,and the conditions ρ(w, z) = ρ(z,w), ρ(∞,∞) = 0. The proof of the nextproposition can be found in A. Beardon’s book ([Be91], Theorem 2.3.2., p.33).

77


Proposition 4.5.4. The transformation T ∶ C ∪ ∞→ C ∪ ∞ given by

Tz = az + bcz + d

with ad−bc = 1 is a Lipschitz function with respect to the chordal metric withLipschitz constant ∥T ∥2 ∶= ∣a∣2 + ∣b∣2 + ∣c∣2 + ∣d∣2.

Proof of Lemma 4.5.3. Let j0,M, l ∈ N be such that A(j0,M, l) ≠ ∅ andj0 ≥ 3. For n ∈ N and a ∈ A(j0,M,n(l + 1)) define

Hn(a) ∶= C(l+1)n(a).Lemma 4.5.3 follows from the First Jarnık Lemma (Lemma 4.2.5) using theclosure of the sets Hn(a) as the family of compact sets once we have estab-lished the existence of some B = B(j0,M, l) > 0 such that for n ∈ N anda ∈ A(j0,M, l + 1)[1..n(l + 1)] and b,c ∈ A(j0,M, l + 1)[1..l + 1]

d(Hn+1(ab),Hn+1(ac)) > B∣Hn(a)∣. (4.15)

First, we show that for some κ = κ(j0,M, l) we have

∀b,c ∈ A(j0,M, l)[1..l + 1] b ≠ c Ô⇒ d(H1(b),H1(c)) > κ. (4.16)

If it happened otherwise, there would be two different a,b in the finite setA(j0,M, l)[1..l+1] such that d(H1(a),H1(b)) = 0. We claim that

∀z ∈ Cl(a) ∀ε > 0 d(z,H1(b)) < ε Ô⇒ d(z,FrH1(a)) ≤ ε. (4.17)

To see it, take z ∈ H1(a), ε, ε′ > 0, w′ ∈ H1(b), and w ∈ H1(b) we haved(z,w) < ε+ε′. By continuity of the function f1 ∶ [0,1]→ C, f(t) = z+t(w−z),and the connectedness of [0,1], f1(t′) ∈ FrH1(a) for some 0 < t′ < 1 . Thesame argument holds when H1(a) and H1(b) are replaced by two cylindersof the same depth.

Call ((pn)n≥0, (qn)n≥0) the Q-pair of a. Let f be the Mobius transforma-tion extending T l∣Cl(a) to the entire plane. By uniform continuity of f onH1(a) (its pole is pl−1

ql−1 /∈ Cl(a)), for every (z,w) ∈ Cl(a) × FrCl(a)∀δ > 0 ∃δ′ > 0 d(z,w) < δ′ Ô⇒ d(T l(z), f(w)) < δ.

Thus, by the continuity of the complex inversion ι, for any z ∈ H1(a) thereis some δ > 0 such that

∀w ∈ C∗ d(T lz,w) < δ Ô⇒ d((T l(z))−1,w−1) < 1

4.

78

Hausdorff Dimension Theory 4.5 Independent Blocks

Therefore, if z ∈H1(a) and w ∈ FrH1(a) could be arbitrarily close, we wouldconclude

d ((T l(z))−1,Fr ι[f[Cl(a)]]) < 1

4.

As a consequence, we would obtain the contradiction

al+1 = [ 1

T l(z)] ∈ a ∈ Z[i] ∶ ∣Ra∣ = 1 ∨ ∣Ia∣ = 1 ∨ ∣a∣ ≤ √8 .

Therefore, (4.16) must hold.Now, take n ∈ N, a ∈ A(j0,M, l)[1..n(l + 1)] with Q-pair (pn)n≥0, (qn)n≥0

and b,c ∈ A(j0,M, l)[1..l+1]. Let R be the Mobius transformation extendingthe inverse of T n(l+1) restricted to Cn(l+1)(a):

R(z) = pn(l+1)z + pn(l+1)−1

qn(l+1)z + qn(l+1)−1

;

then,

d(Hn+1(ab),Hn+1(ac)) = d(R[H1(b)],R[H1(c)])≥ 1

2ρ(R[H1(b)],R[H1(c)])

≥ 1

2∥R∥2ρ(R[H1(b)],R[H1(c)])

≥ 1

2∥R∥2d(R[H1(b)],R[H1(c)]).

As a consequence, for a constant depending on j0,M, l (κ1),

d(Hn+1(ab),Hn+1(ac)) ≫ 1∥R∥2≥ ∣Cn(l+1)(a)∣ = ∣Hn(a)∣.

An immediate outcome of the Isodiametric Inequality (Theorem 1 in[EvGa15], p. 69) and Lakein’s First Theorem from Chapter 1 is the fol-lowing lemma.

Lemma 4.5.5. There exist absolute constants such that for any regular se-quence a ∈ ΩHCF ∀n ∈ N mCn(a) ≍ ∣Cn(a)∣2.

On the basis of equation (3.2) in Chapter 3, and an inductive argument,we can deduce the following lemma.

79


Lemma 4.5.6. For any j0 ∈ N and sufficiently large m,M there exists K =K(j0,m,M) > 0 of the form K = Rmr such that

1. r = r(j0) > 0 and R = R(M) 1 when M →∞ for every j0 > 0.

2. For any k ∈ N and any a ∈ A(j0,m,M)[1..r(m + 1)] we have

∑c

mC(k+1)(m+1)(a;c) ≥KmCk(m+1)(a)where the sum runs over A(j0,m,M)[1..m + 1].

4.5.2 Proof of Theorem 4.4.3

Proof of Theorem 4.4.3. Fix j0,M, l ∈ N such that A(j0,M, l) ≠ ∅. Definek1 ∶= k1(l,M) by

k1(l,M) ∶= (2 − 1

M + 1)−4

φ−4(l+1),and take the K =K(l,M) provided by Lemma 4.5.6; hence,

∀t ∈ N k1mCt(l+1)(a) ≥ mC(t+1)(l+1)(a;c),∀t ∈ N ∑c

mC(t+1)(l+1)(a;c) ≥KmCt(l+1)(a),where c runs along A(j0,M, l)[1..l+1]. Let 0 < α < β be the implied constantsin Lemma 4.5.5, then

∑c

∣C(k+1)(l+1)(a;c)∣2s ≥ αs∑c

(mC(k+1)(l+1)(a;c))s= αs∑

c

mC(k+1)(l+1)(a;c) 1

(mC(k+1)(l+1)(a;c))1−s

≥ αs 1

k1−s1 (mCk(l+1)(a;c))1−s∑

c

mC(k+1)(l+1)(a;c)≥ αs K

k1−s1 (mCk(l+1)(a;c))1−smCk(l+1)(a;c)

= αs Kk1−s

1

mCk(l+1)(a;c)s≥ (α

β)s K

k1−s1

∣Ck(l+1)(a;c)∣2s .By Lemma 4.5.3, dimH F(j0,M, l) ≥ 2s whenever

γsK

k1−s1

≥ 1,

80

Hausdorff Dimension Theory 4.6 Notes and comments

where γ = αβ−1, which holds if and only if

s log γ + logK − (1 − s) log k1 ≥ 0 ⇐⇒ (1 − s) ≥ logK + log γ

log k1 + log γ> 0.

Thus, it suffices to show that the last expression can be as small as we please.Expanding k1 and K, keeping the notation of Lemma 4.5.6, we get

0 < logK + log γ

log k1 + log γ= log (R(M)) + log r+log γ

l+1

logφ + log γl+1

,

which can be arbitrarily small for suitable l,M . Hence, by taking s → 1 weconclude that

supM,l∈NdimH F(j0,M, l) = 2.

4.6 Notes and comments

1. Theorem 4.5.2 is new. The omitted details are part of a paper in process.

81

4.6 Notes and comments Hausdorff Dimension Theory

82

Chapter 5

Hurwitz - Tanaka ContinuedFractions

In this chapter, we discuss the ergodic theory of the continued fraction algo-rithm suggested by Julius Hurwitz (Adolph Hurwitz’s older brother) in 1895and re-discovered by S. Tanaka about a century later ([Ta85]). We definethe Hurwitz-Tanaka Continued Fractions within the framework of Dani andNogueira and see that they provide a counterexample to Theorem 1.1.2. Amore dramatic feature of the Hurwitz-Tanaka algorithm is that the resultingcontinued fractions might not even make sense, let alone represent the origi-nal complex number. However, discarding a suitable null set, a satisfactoryergodic theory can be developed.

Notation. The ideal of Z[i] generated by 1 + i is (1 + i). The set of non-zero elements of (1+ i) is (1+ i)∗. The complex inversion z ↦ z−1 is denotedby ι. ∨ is the logic disjunction connector. If A is a finite set and B ⊆ AN,then σ ∶ B → AN is the shift map σ(x1, x2, x3, . . .) = (x2, x3, . . .). The samenotation σ is used for two-sided sequences.

Main references. For the ergodic theoretical aspects of the Hurwitz-Tanakacontinued fractions, see [Ta85]. For more details and historical aspects, see[Os16].

5.1 Definition and convergence

Identify R2 with C in the natural way: x + iy ∼ (x, y). Let Λ be the latticegenerated by α ∶= (1,1) ∼ 1 + i and α ∶= (1,−1) ∼ 1 − i. Since the squarerα+ sα ∶ 0 ≤ r, s < 1 is a complete set of representatives of R2/Λ, so it is its

83

5.1 Definition and convergence Hurwitz - Tanaka Continued Fractions

translate

T ∶= rα + sα ∶ −1

2≤ r, s < 1

2 .

With the partition a + T ∶ a ∈ (1 + i) we can naturally define a choicefunction fT ∶ C→ (1 + i) ⊆ Z[i] by the condition

∀z ∈ T z ∈ fT (z) +T.

and the corresponding fT -Gauss map by τ ∶ T∗ → T∗, τ(z) = z−1 − fT (z−1).Definition 5.1.1. The Hurwitz-Tanaka continued fraction (TCF) ofa complex number z is the fT -sequence of z. If (an)n≥0 is the TCF of z, weadopt the formal notation [0;a1, a2, . . .]T .

−1 1

−1

1

0

−3 −2 −1 1 2 3

−3

−2

−1

1

2

3

0

Figure 5.1: Left: T. Right: Inverse of T.

Straight forward computations tell us that the TCF of −1 is the constantsequence (0; 0,0, . . .). Clearly, [0; 0,0, . . .]T cannot converge to −1. It is natu-ral to ask under what conditions is the TCF of a complex rational convergentor even finite. J. Hurwitz asked this to himself too and found the followinganswer.

Theorem 5.1.1 (J. Hurwitz, 1895). A number z ∈ Q(i) has infinite TCF ifand only if τn(z) = −1 for some n ∈ N0; moreover,

⋃n≥0

τ−n[−1] = hk∈ Q(i) ∶ h, k ∈ Z[i] h ≡ k ≡ 1 (mod 1 + i) . (5.1)

Proof. Clearly, the TCF of a complex rational z ∈ T is infinite when τn(z) =−1 for some n. Assume that z = hk ∈ T is rational, h, k ∈ Z[i] co-prime, and

84

Hurwitz - Tanaka Continued Fractions 5.1 Definition and convergence

that τn ≠ −1 for all n ∈ N0. For some co-prime h1, k1 ∈ Z[i] and some a ∈ (1+i)we have

h1

k1

= τ (hk) = k

h− a ∈ T, (5.2)

and ∣h1∣ ≤ ∣k1∣. If the equality held, then we would have h1k1

= −1; hence,∣h1∣ < ∣k1∣ = ∣h∣. Repeating the argument we eventually get hn = 0, thealgorithm terminates, and the TCF is finite. This shows the first assertion.

We start proving the second part by noting that for h, k ∈ Z[i] co-prime,h1, k1 co-prime, and a ∈ (1 + i) such that (5.2) is true, we have that

hh1 ≡ kk1 (mod 1 + i). (5.3)

Let us start with ⊆. Take z = hk ∈ T ∩Q(i) with h, k ∈ Z[i] co-prime. If

τ(z) = −1, then h1 = −k1 and k ≡ −h ≡ h (mod 1+ i), so h ≡ k ≡ 1 (mod 1+ i)(otherwise h, k ∈ (1 + i)). A recursive argument now shows that h ≡ k ≡ 1(mod 1 + i) if τn(hk) = −1 for some n ∈ N0.

To prove ⊇, let z ∈ T∖⋃n≥0 T −n[−1] be a complex rational, then TN(z) = 0

for some N and z = pN (z)qN (z) . It follows directly from the definition of Q-pair

that

∀n ∈ N0 p2n+1 ≡ q2n ≡ 1 (mod 1 + i), p2n ≡ q2n+1 ≡ 0 (mod 1 + i).Therefore, z is not the quotient of h, k ∈ Z[i] satisfying h ≡ k ≡ 1 (mod 1 +i).

The convergence situation is by far simpler in the irrational case. Indeed,take z ∈ T′. Since T ∩ C1(0) = −1, the orbit of z under τ , (τn(z))n≥0, isentirely contained in the unit disc. Therefore, the iteration sequence of z isnon-degenerate (cfr. Theorem 1.1.2).

Theorem 5.1.2. If z is any number in T′ and (an)n≥0 its TCF, then z =[a0;a1, . . .]T .

We shall use freely the notions of cylinders, maximal feasible sets, regularand irregular objects. They are defined just as in the HCF context. Theset of admissible sequences (for irrational numbers) is denoted by ΩTCF andwe define ΩTCF(n), ΩTCF

R as in Chapter 3. The set of regular sequencesis denoted by ΩTCF

R and the set of irregular sequences by ΩTCFI . If b is a

sequence in (1 + i)∗, CTn (b) is the cylinder of depth n.The next lemma can be shown using the dual algorithm (defined below)

(see [Ta85]) or with an inductive argument (see [Os16]). Note that the ex-ponent of ∣qn∣ is −1, not −2.

85

5.2 Laws of Succession Hurwitz - Tanaka Continued Fractions

−1 1

−1

1

0

1− i

2

1 + i

2 + 2i

2i

−1 + i

−2

−1− i

−2− 2i

−2i

2− 2i

−2 + 2i

Figure 5.2: Partition of T after a1

Lemma 5.1.3. Let z = [0; b1, b2, . . .] ∈ T′ have Q-pair (pn)n≥0, (qn)n≥0, then

1. (∣qn∣)n≥1 is strictly increasing,

2. For all n ∈ N we have ∣z − pnqn

∣ < ∣qn∣−1,

3. For all n ∈ N we have ∣CTn (b)∣ ≪ ∣qn∣−1.

5.2 Laws of Succession

The structure of the space ΩTCF is simpler than the one of ΩHCF. To see it,let us start by writing

∀b ∈ (1 + i) T(b) ∶= τ[C1(b)] = −b + (ι[T] ∩ (b +T)).Thus, T(b) = T whenever ∣b∣ ≥ 2 and, except for the boundaries, the setsT(1 + i),T(−1 + i),T(−1 − i),T(1 − i) are rotations of each other (see Figure5.3). We also have T(0) = −1, so

ΩTCFR ⊆ (an)n≥0 ∶ ∀n ∈ N an ∈ (1 + i)∗.

Note now that for any b ∈ ΩTCFR (n), b = (b1, . . . , bn), we have

τ[Cn(b1, . . . , bn)] = Cn−1(b2, . . . , bn) ∩T(b1)86

Hurwitz - Tanaka Continued Fractions 5.2 Laws of Succession

T(b), |b| ≥ 2 T(1 + i) T(1− i) T(−1− i) T(−1 + i)

Figure 5.3: Maximal feasible regions

(the first term in the right is omitted when n = 1). Therefore, whenever∣b1∣ ≥ 2,

τ[Cn(b1, . . . , bn)] = Cn−1(b2, . . . , bn) ∩T == Cn−1(b2, . . . , bn). (5.4)

Assume now that ∣b∣ = √2, b = −1 − i say (the other possibilities are treated

similarly). From

T(−1 − i)−1 = T−1 ∩ z ∈ C ∶ Iz ≥ −1 +Rzwe obtain for every b ∈ (1 + i)∗ that

T(−1 − i)−1 ∩ (b +T(b)) = b +T(b) ∨ m(T(−1 − i)−1 ∩ (b +T(b))) = 0.

Therefore, when (b1, b2) ∈ ΩTCFR (2) we have τ[C2(b1, b2)] = C1(b2). As a conse-

jklmn

pqrst

a

b

f1

g1

h1

i1

−3 −2 −1 1 2

−2

−1

1

2

3

0

Figure 5.4: The region T(−1 − i)−1

quence, since τ restricted to any cylinder is injective, for all b = (b1, . . . , bn) ∈ΩTCFR (n) we have

τ[Cn(b)] = τ[Cn(b) ∩ C1(b1)]= Cn−1(b2, . . . , bn) ∩T(b1) ∩ C1(b2)= Cn−1(b2, . . . , bn) ∩ C1(b2)= Cn−1(b2, . . . , bn).87

5.3 The Ergodic Theory Hurwitz - Tanaka Continued Fractions

A recursive application of the previous identities yields

∀n ∈ N ∀b ∈ ΩTCFR τn[Cn(b)] = T(bn).

Thus, given b ∈ ΩTCFR (n) and b ∈ (1+i)∗ the membership of bb to ΩTCF

R (n+1)depends on bn and b.

We summarize the discussion in the next result.

Theorem 5.2.1. The dynamical system (ΩTCFR , σ) is a Markov shift.

Call M the infinite matrix describing ΩTCFR , M ∶ (1+i)∗×(1+i)∗ → 0,1.

Put J = z ∈ (1 + i)∗ ∶ Iz ≥ −1 +Rz. Then M is given by

∀b, c ∈ (1 + i)∗ M(b, c) =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1, if ∣b∣ ≥ 2,

1, if b = ir(−1 − i), c ∈ irJ , r ∈ [0..4]0, if b = ir(−1 − i), c /∈ irJ , r ∈ [0..4].

Irregular objects. Let us point out that irregular numbers do exist. In-deed, the segments (−1 − i,mα + α) for any m ∈ Z ∖ 0 are irregular (seeFigure 5.4), as well as (−1 + i, α +mα) for any m ∈ Z ∖ 0. We leave to thereader to show that mTI = 0 and dimH T = 1.

5.3 The Ergodic Theory

The simpler structure of the associated shift space allowed S. Tanaka toconstruct the natural extension of (T′, τ) and even to obtain a measure νequivalent to m making (T,B(T), τ, ν) ergodic.

5.3.1 The Dual Algorithm

Recall that the natural extension of a one-sided shift space is a two-sidedshift space (cfr. Appendix A). Then, in order to build the natural extensionof (ΩTCF

R , σ) we must determine the space of sequences (bn)n≥1 in (1 + i)∗such that ∀n ∈ N bnbn−1 . . . b1 ∈ ΩTCF

R (n).From a symbolic point of view, it is not hard to determine the desired se-quence space, for we already know the matrix determining ΩTCF. Moreover,it can be shown that the space is also described by an infinite matrix M . Forexample, if b ∈ (1 + i)∗ satisfies Ib > 1 + ∣Rb∣, then

∀c ∈ (1 + i)∗ M(b, c) = ⎧⎪⎪⎨⎪⎪⎩0, c ∈ −1 + i,1 + i,1, otherwise.

88

Hurwitz - Tanaka Continued Fractions 5.3 The Ergodic Theory

It turns out that the sequence space we are looking for is also generated by acomplex continued fractions algorithm included in the framework of ChapterI. We refer to this process as the dual algorithm.

The dual algorithm can be defined with a suitable partition of C. First,consider the sets

S0 ∶= D(0),S1 ∶= D(0) ∖D(−1 − i), S2 ∶= iS1, S3 ∶= −S1, S4 = −iS1,

S5 ∶= D(0) ∖ (D(−1 + i) ∪D(−1 − i)) , S6 ∶= iS5, S7 ∶= −S5, S8 ∶= −iS5.

Rather than describe meticulously the translates of the sets Sj partition-ing C, we content ourselves with Figure (5.6). To each a ∈ (1 + i) denoteby S(a) the set Sj such that a +S(a) belongs to the partition. The choicefunction fS ∶ C → (1 + i) is given by the condition z ∈ fS(a) +S(a) and thecorresponding fS-Gauss map by S ∶S∗ →S, S(z) = z−1 − fS(z−1).

The partition of S induced by the cylinders of depth 1 is depicted inFigure 5.5.

−1 1

−1

1

0

Figure 5.5: Partition of S1 after a1(z).From now on, we exclude the null set ⋃n≥0 S−n[C(0; 1)] and denote by

∆TCF the resulting sequence space. Our omission forbids the appearances of0 as a term of any sequence in ∆TCF. Based on the geometry of the dualalgorithm, it can be shown that (∆TCF, Td) is precisely the sequence spacethat we have been after.

Theorem 5.3.1. Let b = (bn)n≥1 take values in (1 + i)∗. Then, b ∈ ∆TCF ifand only bn+1bn ∈ ΩTCF(2) for every n.

89


−3 −2 −1 1 2 3

−3

−2

−1

1

2

3

0

Figure 5.6: Partition of C for the dual algorithm.

5.3.2 The Natural Extension

The space underlying the natural extension of (IR, τ) (with the suitableergodic measure) is given by

Z = (w, z) ∈S ×T ∶ b1(w)a1(z) ∈ ΩTCF(2) ,R ∶ Z → Z ∀(w, z) ∈ Z R(w, z) = ( 1

a1(z) +w, τ(z)) .Exploring the condition b1(w)a1(z) ∈ ΩTCF(2) we can actually write

Z = 8⋃j=1

Sj ×Xj,

where X1, . . . ,X8 is the partition of T shown in Figure 5.7 (the involved arcshave their center in α

2 , iα2 ,−α2 ,−iα2 and radius 2−1/2).

Theorem 5.3.2 (S. Tanaka, 1985). Let h ∶ Z → R>0 be given by

∀(w, z) ∈ Z h(w, z) = 1∣1 +wz∣4 .Then, the measure ρ, dρ = hdm, is finite and R-invariant.

The Theorem of Change of Variable plays an important role again. In-deed, the R invariance follows from the equality

∀(w, z) ∈ Z ∣DR(w, z)∣h(R(w, z)) = h(w, z),90

Hurwitz - Tanaka Continued Fractions 5.3 The Ergodic Theory

X1

X2

X3X4X5

X6

X7

X8

−1 1

−1

1

0

Figure 5.7: Partition of T = ⋃jXj

where DR is the Jacobian of R (regarded as a map from a subset of R4 intoitself). Although harder to prove, the finiteness of the measure verified bybounding separately each term of the sum

ρ1(Z) = ∫Zh(w, z)dm(w, z) = 8∑

j=1∬

Sj×Xj h(w, z)dm(w)dm(z).5.3.3 The Ergodic Measure

Let π2 ∶ Z → T denote the projection on the second coordinate, i.e. π(w, z) =z. Since π R = τ π, the measure ν given by

∀B ∈B(T) ν(B) ∶= ρ(π−1[B])is τ -invariant. The irreducibility of ν— and thus its ergodicity— is provenessentially the same way as for HCF. Some care must be taken, though.Take z = [0; b1, b2, . . .]T ∈ T′ with Q-pair (pn)n≥0, (qn)n≥0. The inverse of therestriction of τn to CTn (b) is

ψb,n(z) = pn−1z + pnqn−1z + qn , ∣ψb,n(z)∣ = 1∣qn−1z + qn∣2 = 1

∣qn∣2 ∣1 + z qn−1qn∣ .

Since ∣z∣ can be arbitrarily close to 1, it is not entirely clear that a Renyi-typecondition (such as (3.1)) holds. However, 3∣qn−1(z)∣ ≤ 2∣qn(z)∣ is true for alln ∈ N almost everywhere (with respect to m).

91


Proposition 5.3.3. For almost every z = [0; b1, b2, b3, . . .]T ∈ T we have that

∀n ∈ N supz∈T(b,n) ∣ψ′b,n(z)∣2 ≪ inf

z∈T(b,n) ∣ψ′b,n(z)∣2 , (5.5)

where T(b, n) = τn[CTn (b)].From the explicit formula for the density of ρ and Z = ⋃jSj × Xj, S.

Tanaka could find an explicit formula for the density of ν. First, define themap A ∶ R2>0 → R by

A(r, s) = r2 arctans

r+ s2 arctan

r

s− rs.

Considering α1 = α, α2 = α, α3 = −α, and α4 = −α define the functionsf1, f2, f3, f4 by

f0(z) ∶= 1

ρ1(Z) π(1 − ∣z∣2)2,

∀j ∈ [1..4] fj(z) ∶= 1

ρ1(Z)A( 1

1 − ∣z∣2 , 1∣z − αj ∣2 − 1) .

Theorem 5.3.4. The measure ν in (T, τ) is ergodic and equivalent to theLebesgue’s measure. The density function of ν, f , is given by

f(z) =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

f0(z) − fj(z) if z ∈Xj for j ∈ [1..4],f0(z) − f1(z) − f2(z) if z ∈X5,

f0(z) − f2(z) − f3(z) if z ∈X6,

f0(z) − f3(z) − f4(z) if z ∈X7,

f0(z) − f4(z) − f1(z) if z ∈X8.

A Khinchin-Levy type result and the entropy of (IR, τ, ν) can be obtainedwith the arguments used for HCF.

Theorem 5.3.5. Let k ∶ T→ R be given by k(z) = log ∣z∣, then k ∈ L1(ν) andfor almost every z = [0; b1, b2, . . .] ∈ T we have

limn→∞ lim

n→∞log ∣qn∣n

= −∥k∥1.

Moreover, the entropy hν(τ) of the system (T,B(T), τ, ν) is hν(τ) = 4∥k∥1.

Similar results also hold for the dual system (S′, S).92

Hurwitz - Tanaka Continued Fractions 5.4 Notes and Comments


1. S. Tanaka defined in a slightly different way theQ-pair ((hn)n≥0, (kn)n≥0)of his continued fractions. While the recurrences are the same, his ini-tial conditions are

h−2 = 0, h−1 = α, k−2 = α, k−1 = 0.

If ((pn)n≥0, (qn)n≥0) is the usual Q-pair, we have αpn = hn and αqn = knfor every n. The discrepancy is irrelevant. We changed the notation ofS. Nakada’s paper aiming for consistency.

2. As noted before, Hurwitz - Tanaka continued fractions are a particularcase of Iwasawa continued fractions. Thus, the general ergodic theorydeveloped by A. Lukyanenko and J. Vandehey, although they provideno calculations of the entropy.

3. Our proof of Theorem 5.1.1 seems to be new. It is much simpler thanthe ones given in [Os16].

4. We have made no attempt in working with the approximation proper-ties of the Hurwitz-Tanaka continued fractions, because the approxi-mation rate is worse than HCF. This is not surprising, for the set wherethe elements can belong is significantly reduced.

93

5.4 Notes and Comments Hurwitz - Tanaka Continued Fractions

94

Chapter 6

Current and Future Research

In this brief chapter, we discuss some problems related to Hurwitz. The firsttwo are the subject of ongoing research.

Notation. If µ is a measure on a subset of Rn, then µ is its Fourier trans-form.

References. We add some references to each problem separately.

6.1 Current Research

6.1.1 A Kaufman Measure in the Complex Plane

In 1980, R. Kaufman published the construction of a measure supported in aparticular compact set with a polynomially decreasing Fourier transform. Hisresult has had implications, for example, in research related to the LittlewoodConjecture (cfr. [HJK]). Some hypothesis imposed by R. Kaufman were laterweakened by M. Queffelec and O. Ramare in [QuRa03].

For each N ∈ N let FN be the set of real numbers in [0,1] whose regularcontinued fraction is bounded by N .

Theorem 6.1.1 (R. Kaufman, 1980). Let N ∈ N be such that dimH FN > 12 .

Then, FN supports a probability measure λ = λN such that for some η > 0 wehave ∀u ∈ R ∣λ(u)∣ ≪N ∣u∣−η.

We can split R. Kaufman’s proof into two parts. First, the measure isbuilt. Then, the Fourier transform conditions are verified. We give a verybroad overview of both steps.

95

6.1 Current Research Current and Future Research

Construction. Let N be as in the statement. We can naturally iden-tify FN with [1..N]N. Using the Hausdorff dimension hypothesis, Kaufmandefined a probability measure µ on [1..N]n. A sequence of independent iden-tically distributed random variables (fn)n≥1 with finite mean is defined onthis space. By the Weak Law of Large numbers, for large N the probabilityof N−1(f1(a1) + . . . + fN(aN)) being close to the mean of f1 is greater than12 . The probability measure λ is defined by conditioning µN to this set andextending it to the countable product.

Bounding the Fourier transform. By construction, λ can be writtenas λ = ν ⊗ λ. Then, by applying Fubini’s Theorem and relating λ with theLebesgue measure, Kaufman reduced the problem to bound some oscillatoryintegrals in one dimension.

The complex analogue. We express our first question as follows:

Problem 1. Is there an analogue Kaufman-like measure for complex plane?

We decided to adapt Kaufman’s construction using Hurwitz ContinuedFractions. Although a significant part of the work is already done, the extradimension poses new difficulties. The first important problem is encounteredin the construction. The (HCF) Laws of Succession make an straight forwardadaptation of Kaufman’s argument impossible. However, by Theorem 4.5.2we can obtain sets with Hausdorff Dimension arbitrarily close to 2 formed bynumbers whose HCF can be understood as a sequence of independent blocks.

The multidimensional treatment of oscillatory integrals is considerablyharder than the one-dimensional. In our case, the complications come fromobtaining a satisfactory lower bound of

z + qN(a) + qN(b)qN−1(b) ,

where z ∈ F, and a,b are valid segments. For regular continued fractions theprevious expression can be easily bounded, because x ∈ (0,1) and continuantsare natural numbers and grow exponentially.

For details on the real Kaufman’s measure, see [K80], [QuRa03], and[JoSa16]. The complex Kaufman measure is a work in progress.

6.1.2 A Generalization of Good’s Theorem

In view of the successful extension of Good’s Theorem to HCF, we can askfor a generalizations.

96

Current and Future Research 6.2 Future Research

Problem 2. Can Good’s Theorem be extended to Schweiger Fibred Systems?

We must consider only those Schweiger Fibred Systems with infinitelymany digits. Moreover, we have to define how to interpret an → ∞ for theregular continued fractions and ∣an∣ →∞ for HCF. We suggest to work withfunctions δ ∶ I → R>0 such that for any M > 0 the inverse image of [0,M]under δ is finite. On the basis of the proofs of the real and complex versions ofGood’s theorems, we suspect that some growth conditions must be imposedon δ.

In both versions of Good’s Theorem irregular elements can be overlooked.This is unlikely to be the case on higher dimensions. For example, if weextend directly the construction of HCF to three dimensions (changing thecomplex inversion for the inversion with respect to the unit sphere), thenthe corresponding irregular numbers might form a set of dimension 3− 1 = 2.Thus, we might have to consider separately regular and irregular numbers.

Rather than obtaining that the Hausdorff dimension of a set is half ofthe ambient space’s, we expect to obtain only estimates. We suspect thatirregular elements are the key factor that will allow the Hausdorff dimensionof the set of interest to vary.

6.2 Future Research

We now list a few problems that might be of interest.

Problem 3. Can we get rid of the lower bound of the elements in Theorem2.3.2?

Problem 4. Is there a Gauss-Kuzmin theorem for Iwasawa Continued frac-tions?

Problem 5. Is there a Borel-Bernstein Theorem for HCF?

A. Nogueira gave in [No01] a Borel-Bernstein Theorem for a family mul-tidimensional continued fractions (defined in the geometric way). However,it is not entirely clear that his work includes HCF.

Problem 6. Is (F, T, µ) an α, ρ-mixing system?

The previous problem was posed by Dr. Poj Lertchoosakul.

Problem 7. Are there numbers ζ ∈ C ∖ Q(i) with an automatic HCF andsuch that ∣ζ ∣2 = n ∈ N?

97

6.2 Future Research Current and Future Research

98

Appendix A

In this appendix, we recall some elementary formulæon the inversion of circlesin the complex plane. The proofs of the propositions below can be safelyomitted, for they are just elementary exercises.

Inversion Formulæ

Let ι ∶ C∗ → C∗ be the complex inversion; i.e. ι(z) = z−1 for all z ∈ C∗. Forany z0 ∈ C and any ρ > 0 let C(z0, ρ) be the circle of radius ρ and center z0:

C(z0, ρ) = w ∈ C ∶ ∣z −w∣ = ρ.Proposition (Inversion of lines and circles).

1. Let z0 ∈ C and ρ > 0. Then,

ι[C(z0, ρ)] = ⎧⎪⎪⎨⎪⎪⎩C ( z0∣z0∣2−ρ2 , ρ∣ρ2−∣z0∣2∣) if ρ ≠ ∣z0∣L ∶ 2R(zz0) = 1 if ρ = ∣z0∣,

where L ∶ 2R(zz0) = 1 is the line z ∈ C ∶ 2R(zz0) = 1.

2. Let a, b ∈ R, (a, b) ≠ (0,0) and consider the lines La,b, L′a,bLa,b ∶ ax + by = 1, L′a,b ∶ ax + by = 0.

Then,

ι[La,b] = C (a2− i b

2,

√a2 + b2

2) , ι[L′a,b] = z = u + iv ∶ au − bv = 0.

99

Appendix A

100

Appendix BErgodic Theory

In this appendix, we establish notation regarding ergodic theory. Althoughmost proofs are omitted, we do provide the appropriate references.

Definitions

Let X = (X,M) be a measurable space and τ ∶X →X a measurable function.

i. Let µ be a measure on X. The function τ is measure preserving oran endomorphism if

∀E ∈M µ(E) = µ(τ−1[E]).The quadruplet (X,M, µ, τ) is a measure preserving system andwhen µ is a probability measure, it is a probability measure pre-serving system.

ii. A measure µ in X is irreducible if

∀E ∈M τ−1[E] = E Ô⇒ µE = 0 ∨ µ(X ∖E) = 0.

iii. The system (X,M, µ, τ) is ergodic if it is a measure preserving systemand τ is irreducible. If the measurable space (X,M) and the measurablemap τ are given, a measure µ is ergodic if the system (X,M, µ, τ) isergodic.

iv. The function τ is non-singular if

∀E ∈M µ(τ−1[E]) = 0 Ô⇒ µ(E) = 0.

Let (X,M, µ, τ) be a (probability) measure preserving system.

101


1. τ is weak-mixing if

∀A,B ∈M limn→∞

1

n

n−1∑i=0

∣µ(τ−i[A] ∩B) − µAµB∣ = 0.

2. τ is strong-mixing if

∀A,B ∈M limn→∞µ(τ−i[A] ∩B) = µAµB∣ = 0.

Ergodic Theorems

The following is a fundamental result in ergodic theory. Its proof can befound, for example, in the Walters’ monograph [Wa82] as Theorem 1.14,p.34.

Theorem (Birkhoff Ergodic Theorem). Suppose (X,B, µ, T ) is a finite orσ-finite measure preserving system. Then, for every f ∈ L1(µ) there exists afunction f∗ ∈ L1(µ) such that f∗ T = f∗ and

limn→∞

1

n

n−1∑i=0

f T i = f∗holds µ-almost everywhere. If µX < +∞, then ∫ fdµ = ∫ f∗dµ. In particular,if µ is a probability measure and T is ergodic, then f∗ = ∫ fdµ.

An application of Birkhoff’s Ergodic Theorem to indicator functions givesthe following corollary.

Corollary. Let (X,B) be a measurable space and T ∶ X → X a measurablemap. If µ and ν are probability measures such that both systems (X,B, µ, T )and (X,B, ν, T ) are ergodic, and ν ≪ µ, then µ = ν.

Partitions

Let (X,M, µ) be a probability space. By a partition we mean an at mostcountable collection Ann ⊆M such that

∀n,m ∈ N, µ(An ∩Am) = δn,m, µ(X ∖⋃nAn) = 0,

where δn,m is the Kronecker symbol. Let P be a partition of X. For x ∈ X,P(x) ∈ P is an element of the partition such that x ∈ P(x). It poses no threat

102


to us that x ↦ P(x) is uniquely defined only µ-almost everywhere. If X isalso a metric space, the diameter of P is

diamP ∶= sup∣P ∣ ∶ P ∈ P.We define a partial order on the partitions of X by writing P ≤ Q for twopartitions P, Q whenever each member of P is the union (up to a µ-null set)of elements of Q. The sum of P and Q is

P ∨Q = P ∩Q ∶ P ∈ P,Q ∈ Q .Note that P,Q ≤ P ∨ Q and that the sum is minimal with respect to thisproperty. If Pnn is an at most countable family of partitions, we define

⋁n∈NPn

to be the minimal partition P such that Pn ≤ P for every n.Suppose that τ ∶ X → X is an endomorphism. For any partition P we

define τ−1[P] = τ−1[P ] ∶ P ∈ P and

P−τ ∶= ∞⋁k=0

τ−k[P],where τ−(n+1)[P] ∶= τ−1[τ−n[P]]. If τ is invertible (almost everywhere) withmeasurable inverse, we can similarly define

P+τ ∶= ⋁k∈N τ

kP.A measurable partition P is a generator of τ if P−τ = ε, where ε is the trivialpartition ε = X. If τ is an automorphism, we say that P is exhaustive ifP+τ = ε and τ−1[P] ≤ P.

When τ has been fixed, we define for P the sequence (Pn)n≥1 by

∀n ∈ N Pn ∶= n⋁k=0

τ−kP,Entropy

The notion of entropy, broadly, speaking tells us about the uncertainty of anexperiment. A nice discussion can be found in [OlVi16].

Let (X,M, µ) be a finite measure space. The entropy of a partition Pis

Hµ(P) = ∑P ∈P µP log(µP ).

103

6.3 Rokhlin’s Natural ExtensionAppendix B

Ergodic Theory

It can be shown that n ↦ Hµ(Pn) is sub-additive, hence the following limitexists:

hµ(τ,P) ∶= limn→∞

1

nHµ(Pn).

The entropy of the system (X,M, µ, τ) is

hµ(τ) ∶= suphµ(f,P) ∶ hµ(τ,P) <∞ .The next theorem appears in [OlVi16] as Theorem 9.3.1, p.262.

Theorem (Shannon - McMillan - Breiman). Let (X,M, µ) be a probabilityspace and let P be an at most countable and measurable partition. Supposethat Hµ(P) < +∞. Then, if τ ∶ X→ X is a measure preserving transformation,for µ almost all x ∈X the following limit exists

hµ(τ,P, x) ∶= − limn→∞

logµ(Pn(x))n

.

Moreover, hµ ∈ L1(µ) and

∫ hµ(τ,P, x)dµ(x) = hµ(τ,P).If (X,M, µ, τ) is ergodic, then

∀µx ∈X hµ(τ,P, x) = hµ(τ,P).The following result can be found as Corollary 9.2.10., p. 257 in [OlVi16].

Theorem 6.2.1. If P is a partition such that diamPn(x)→ 0 as n→∞ forµ almost all x ∈X. Then, hµ(τ) = hµ(τ,P).

6.3 Rokhlin’s Natural Extension

The full proofs of the statements in this sections are in V.A. Rokhlin’s papers[Ro64] and [Ro67].

Let X = (X,B, µ, τ) be a measure preserving system. We say that apartition P is invariant if τ−1[P] ≤ P. Let P be an invariant partition andconsider the quotient space X/P. Let f ∶X →X/P be the factor map givenby f(x) = P(x). We say that Z ⊆X/P is measurable if f−1[Z] ∈B and, inthis case, we write

µ(Z) ∶= µ (f−1[Z]) .We can further consider the factor endomorphism τP ∶ M/P → M/P as themap satisfying f T = TP f .

104

Appendix BErgodic Theory 6.3 Rokhlin’s Natural Extension

Definition. Let X = (X,BX , µX , τ) be a measure preserving system. A mea-sure preserving system X′ = (X ′,B′, µ′, τ ′) is a natural extension of X ifX′ has an exhaustive measurable partition ζ such that τ ′ζ is isomorphic to τ .

Theorem (V. A. Rokhlin, 1961). Any measure preserving system X′ has aunique natural extension X′. Furthermore, X is ergodic if and only if X′ is.

Corollary. The natural extension of a one-sided Bernoulli shift space (overa metric space) is a two-sided Bernoulli shift on the same space of states.

Sketch of proof. We consider a simpler context: assume that E is a com-plete and separable metric space. Let BE be its Borel σ-algebra and µEa Borel probability. By the Daniel-Kolmogorov Theorem, the system X =(X,BX , µX , σX) where X = EN, σX the right-shift map and the rest ofthe objects are the usual products is well defined. Similarly, consider Y =(Y,BY , µY , σ = σY ) for Y = EZ.

Let π be the projection π ∶ Y → E given by π(xn)n∈Z = x1 and define thepartition P = π−1[x] ∶ x ∈ Y .Note that (xn)n∈Z ∼ (yn)n∈Z with respect to R ∶= P−σY if and only if xn = ynfor all n ∈ N. It can be shown using the definitions that R is exhaustive,σ−1[R] =R and that σR and σX are isomorphic.

Rokhlin’s natural extension has an important universal property. Namely,it factors every invertible extension of X. That is, if Y = (Y,BY , ν, S) is aninvertible extension of X with factor map ϕ ∶ Y →X, then, there is a measurepreserving map ϕ ∶ Y →X ′ such that ϕ = πϕ.

105

6.3 Rokhlin’s Natural ExtensionAppendix B

Ergodic Theory

106

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112

Papers

113

Purely Periodic and Transcendental Complex Continued

Fractions


Abstract

Adolf Hurwitz proposed in 1887 a continued fraction algorithm for representing complexnumbers. Among other similarities with regular continued fractions, quadratic irrationalscan be characterized in terms of periodic expansions. In this paper we give necessary andsufficient conditions for pure periodicity with respect to Hurwitz algorithm. We also showa complex analogue of a theorem by Y. Bugeaud ([Bu13]) which establishes a combinatorialcondition for transcendence.

1 Introduction

Regular continued fractions are a remarkably useful tool in number theory. The structureof a regular continued fraction expansion sometimes helps to determine algebraic or analyticproperties of the number it represents. A famous result in this direction is the Euler-LagrangeTheorem. It says that an irrational number has an ultimately periodic continued fraction if andonly if it is a quadratic irrational. Another important theorem, due to Liouville, allows us toconstruct transcendental numbers by simply taking continued fractions whose terms grow fastenough. A well known conjecture relating transcendence and boundedness of continued fractionis the following.

Conjecture 1.1 (Folklore Conjecture). If an irrational real algebraic number has a boundedcontinued fraction, then it is a quadratic irrational.

Although the conjecture remains widely open, there are important partial results. On thebase of Roth’s Theorem, Alan Baker showed that whenever the continued fraction of a realirrational α satisfies certain combinatorial condition, then α has to be transcendental. In 1998,Martine Queffelec showed the transcendence of numbers whose regular continued fraction is theThue-Morse sequence over an alphabet a, b ⊆ N. Afterwards, she showed the transcendence ofa larger class of automatic continued fractions. Based on their joint work with Florian Luca onb-ary expansions, Boris Adamczewski and Yann Bugeaud generalized in 2005 some of Queffelec’swork. They showed that palindromic continued fractions and other combinatorial conditionsimply transcendence. These results were crowned by Y. Bugeaud in [Bu13]. The main resultof [Bu13] implies that real numbers with an automatic continued fraction are either quadraticirrationals or transcendental.

In a recent paper ([BuKi15]), Y. Bugeaud and Dong Han Kim defined a function givinganother notion of complexity of an infinite word. With their function, it is possible to state themain theorem of ([Bu13]) in an extremely neat fashion (Theorem 1.1).

We must recall first some definitions.

Definition 1.1. Let A ≠ ∅ be a finite set and a an infinite word on A. The repetitionfunction, r(⋅,a) ∶ N→ N, is

∀n ∈ N r(n,a) = minm ∈ N ∶ ∃i ∈ [1..m − n] ai⋯ai+n−1 = am−n+1⋯am .The repetition exponent of a, rep(a), is

rep(a) ∶= lim infn→∞

r(n,a)n

.

1

1 INTRODUCTION 2

Theorem 1.1 (Bugeaud, 2011). . Let a = a1a2 . . . be a non-periodic infinite word in N. If

rep(a) < +∞,then α = [0;a1, a2, a3, . . .] is transcendental.

By Cobham’s Theorem on the complexity of automatic sequences ([AlSh03], Corollary10.3.2), every automatic sequence has a finite repetition exponent.

Theorem 1.2 (Bugeaud, 2011). Let (an)∞n=1 be an automatic sequence of positive integers. If(an)∞n=1 is not periodic, then α = [0;a1, a2, . . .] is transcendental.

Given the importance of a regular continued fraction, it is natural to look for a similar toolin other contexts. In the complex plane, C, there have been several attempts. An outstandingexample was given by Asmus Schmidt([ASch]). His sophisticated construction focuses on thequality of approximation. A much simpler algorithm was proposed by Adolf Hurwitz ([Hu87]),it is just the straight forward generalization of the nearest integer continued fraction (see Section2 for details). Some other expansions and their associated ergodic theory were studied by JuliusHurwitz, William Leveque, Hitoshi Nakada, Georges Poitu, among others. Lately, some familiesof complex continued fractions have been studied by Shrikrishna Gopalrao Dani and ArnaldoNogueira ([DaNo14], [Da15]).

In this paper, we restrict ourselves to Hurwitz Continued Fractions. While similaritiesbetween Hurwitz and regular continued fractions abound, there important differences. Forexample, Serret’s theorem states that two real numbers belong to the same orbit of PSL(2,Z)/R(acting via Mobius transformations) if and only if the numbers are both rational or if they areboth irrational and the tails of their continued fraction expansion coincide. The analogue fails inC with Hurwitz Continued Fractions. Richard Lakein gave in [La74b] a pair of complex numbers,ζ and ξ, and an element γ ∈ PSL(2,Z[i]) such that ζ = γξ but whose Hurwitz continued fractionsnever coincide.

The most striking results was obtained by Doug Hensley ([He06]) and extended by WiebBosma and David Gruenewald ([BoGu12]). It is a negative answer to the Folklore Conjecturefor complex numbers and Hurwitz Continued Fractions.

Theorem 1.3 (Bosma, W., Gruenewald, D. (2012)). Let n be a natural number. There existsan algebraic number α ∈ C, whose Hurwitz Continued Fraction expansion has bounded elements,such that [Q(i, α),Q(i)] = 2n.

Despite Theorem 1.3, we can also conclude certain properties about the repetition exponentof the continued fractions of algebraic numbers. A trivial consequence of one of our main result,Theorem 5.1, is a weaker analogue of Theorem 1.1.

Theorem 1.4. Let a = (a1, a2, . . .) be the Hurwitz continued fraction of a number ζ ∈ C. If a isnot periodic and √

8 ≤ lim infn→∞ ∣an∣, rep(a) < +∞,

then ζ = [0;a1, a2, . . .] is transcendental.

Corollary 1.5. Let a = (a1, a2, . . .) be the Hurwitz continued fraction of a number ζ ∈ C. If∣an∣ ≥ √8 for n ∈ N, and a is automatic, then ζ = [0;a1, a2, . . .] is quadratic over Q(i) or

transcendental.

In Theorem 3.2 we study purely periodic continued fractions. As in Theorem 1.4 andCorollary 1.5, a lower bound appears. We show that, for our result on purely continued fractions,the lower bound is- while unpleasant- best possible.

Our main results give necessary and sufficient conditions for purely periodic HCF expansions(Theorem 3.2), characterize badly approximable complex numbers (Theorem 4.1), and givenecessary conditions for transcendence of complex numbers (Theorem 5.1).

2 HURWITZ CONTINUED FRACTIONS 3

The paper is organized as follows. In the second part we define properly Hurwitz ContinuedFractions and we discuss the associated shift space. In the third section, we state and provesufficient and necessary conditions on purely periodic continued fractions. In the fourth section,we characterize badly approximable in terms on Hurwitz Continued Fractions (Theorem 4.1).This result was recently shown by Robert Hines ([Hi17]), but our proof is slightly different. Inthe last section we state and show Theorem 5.1, our main transcendence result.

Notation. We will reserve [a0;a1, a2, . . .] for Hurwitz Continued Fractions. We will also needcomplex continued fractions which may fail to be Hurwitz, we represent them by ⟨a0;a1, a2, . . .⟩.By natural numbers, N, we mean the set of positive integers and we write N0 = N ∪ 0. Giventwo functions f, g ∶ N0 → R, the Vinogradov symbol f ≪ g means that for some C > 0 and everyn ∈ N we have f(n) ≤ Cg(n). Finally, for any z ∈ C and A ⊆ C we consider z+A ∶= z+a ∶ a ∈ A.

2 Hurwitz Continued Fractions

Denote by [⋅] ∶ C→ Z[i] the function which assigns to each complex number its nearest Gaussianintegers rounding up to break ties; thus

∀z ∈ C z − [z] ∈ F ∶= z ∈ C ∶ 1

2≤Rz,Iz < 1

2 .

In analogy to the Gauss map, we define the function

T ∶ F∗ → F, T (z) = 1

z− [1

z] ,

where F∗ = F ∖ 0. Writing T 1 ∶= T and Tn+1 ∶= Tn T for n ∈ N we give to any z ∈ C a pair ofsequences of complex numbers (an)n≥0 and (zn)n≥0 through

z0 ∶= z, a0 ∶= [z0],z−1n = Tn−1(z0 − a0), an ∶= [zn],

as long as the iterations of T make sense.Considering a1 as a function from F to Z[i], we get a partition of F induced by the pre-images

of a1 (see Figure 1). It is clear that

K ∶= Clz ∈ F ∶ ∣a1(z)∣ ≥ √8 ⊆ F, (1)

where Cl denotes the closure and , the interior with respect to the usual topology.

Definition 2.1. The Hurwitz Continued Fraction (HCF) of a complex number z is thesequence (an)n≥0 obtained by the above procedure. As defined in [DaNo14], the Q-pair of z isthe pair of sequences (pn)∞n=0, (qn)∞n=0 given by

(p−2 p−1q−2 q−1) = (0 1

1 0) , ∀n ∈ N0 (pn

qn) = (pn−1 pn−2

qn−1 qn−2)(an1) . (2)

The terms of the sequence (pnqn )n≥0 are called the convergents of z.

As expected, the HCF of a complex number ζ is infinite if and only if ζ ∈ C ∖Q(i). In thiscase, we naturally have that the convergents converge to ζ ([DaNo14], [Hu87]).

Not every sequence in Z[i] is the HCF of a complex number. We say that an infinite sequenceof Gaussian integers is valid if it is the HCF of some complex irrational number and we willdenote the set of valid sequences by ΩHCF. By a valid prefix we mean a finite sequence onZ[i] which is the prefix of a valid sequence. A valid segment is defined similarly.

Some necessary conditions for a sequence (an)∞n=0 to belong in ΩHCF follow immediatelyfrom the algorithm.


−0.5 0.5

−0.5

0.5

0

1 − i−1 − i

1 + i−1 + i

−2 − i

−1 − 2i −2i 1 − 2i

2 − i

2

2 + i

1 + 2i

2 + 2i3i

2i

−1 + 2i

−2 + i

−2 −3−3 − i

−3 + i

−2 + 2i−1 + 3i 1 + 3i

3 + i

3

3 − i3 − 2i

2 − 2i−3i

−2 − 2i −1 − 3i

Figure 1: Partition of F.Figure 1: Partition of F induced by a1(z).

Proposition 2.1. Let (an)n≥0 ∈ ΩHCF be the HCF of z. The following inequalities hold

∀n ∈ N ∣zn∣ ≥ √2, ∣an∣ ≥ √

2.

Moreover, we have that

(an)n≥0 ∈ Z[i]N0 ∶ ∀n ∈ N ∣an∣ ≥ √8 ⊆ ΩHCF. (3)

Take z ∈ F∗, then z1 ∈ F−1 ∶= w−1 ∶ w ∈ F (depicted in Figure 3). Suppose that z2 exists.If ∣a1∣ ≥ √

8, then the feasible maximal region for z2 is again F−1. However, if a1 = 1 + i–forexample– z2 belongs to the set

z ∈ F−1 ∶ (1

2≤Rz)&(≤Rz < 1

2) .

In general, the maximal feasible region for z−1n+1 has one of the forms drawn in Figure 2 orone of the rotations obtained by multiplying them times an integral power of i (in the referredfigure we neglect the boundary). Thus, determining whether a sequence is valid or not is morecomplicated than just checking a uniform lower bound.

With a slight notational abuse, we see that (T,ΩHCF) is a shift space which cannot bemodelled as a Markov shift. Indeed, assume there was an infinite the matrix A characterizing1

ΩHCF. On the one hand, the segment 1+2i,−2+2i,1+ i is not valid, so A−2+2i,1+i = 0. However,0,−2+2i,1+i is a valid prefix, which would imply A−2+2i,1+i = 1, a contradiction. We can extend

1In the following sense: a sequence (an)n≥1 in Z[i] belongs to ΩHCF if and only if Aanan+1 = 1 for every n ∈ N.


F1 F2 F3 F4

Figure 2: Feasible regions for z−1n+1.1

0−2 −1 1 2

−2−1

1

2

Figure 3: The set F−1.

this observation to prefixes of arbitrary length. While for any n ∈ N the sequences

(−2 + 2i,2 − 2i, . . . ,2 − 2i,−2 + 2i´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶n repetitions of −2+2i,2−2i

,1 + i),(2 − 2i,−2 + 2i,2 − 2i, . . . ,2 − 2i, ,−2 + 2i´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

n repetitions of −2+2i,2−2i,1 + i) (4)

are valid segments, the sequences

(1 + 2i,−2 + 2i,2 − 2i, . . . ,2 − 2i,−2 + 2i´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶n repetitions of −2+2i,2−2i

,1 + i),(−1 + 2i,2 − 2i,−2 + 2i,2 − 2i, . . . ,2 − 2i, ,−2 + 2i´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

n repetitions of −2+2i,2−2i,1 + i) (5)

are not. Obviously, the conclusion holds if we replace 1 + 2i and −1 + 2i by 2 + i and −2 + i. Wecan obtain more examples using the symmetries of the HCF process.

The lack of Markov structure is a significant difference between the Hurwitz ContinuedFractions and its direct real analogue, the Nearest Integer Continued Fraction (NICF). It canbe easily shown that the set of valid sequences of integers with respect to the NICF algorithmcan be characterized in terms of a transition matrix. This feature complicates the study of finitesequences of the form (aj , aj−1, . . . , a1) where (a1, . . . , aj−1, aj) is a valid prefix. Such sequencesappear naturally since ∀n ∈ N qn+1

qn= ⟨an+1;an, . . . , a1⟩ .

Nevertheless, the denominators (qn)∞n=0 have some useful properties. We borrow the followingresults from [DaNo14] (the third part is not stated in that paper, but it is a direct consequenceof the second part).

Lemma 2.1 (Dani, Nogueira (2011)). Let ζ = [a0;a1, a2, a3, . . .] be an element of C∖Q(i) and(pn)∞n=0, (qn)∞n=0. The following statements hold.

3 PERIODIC CONTINUED FRACTIONS 6

i. The sequence (∣qn∣)n≥0 is strictly increasing.

ii. Set φ = 1+√52 , then

∀n ∈ N0∣qn+1∣∣qn∣ > φ ∨ ∣qn+2∣∣qn+1∣ > φ.

iii. For every n, k ∈ N we have ∣qn+k∣ > φ[ k2]∣qn∣.

In the following, we adopt the notation of [La73]. For every z ∈ C and ρ > 0, D(z, ρ) is theopen disc with radius ρ and centered at z, D(z, ρ) = ClD(z, ρ), and E(z, ρ) = C ∖D(z, ρ).3 Periodic continued fractions

A. Hurwitz proved an analogue to the Euler Lagrange theorem for his continued fractions. Morethan a century later, S.G. Dani and A. Nogueira showed that it still holds for a larger familyof complex continued fractions ([DaNo14]).

Theorem 3.1 (Hurwitz (1887)). Let z be an irrational complex numbers. The Hurwitz Con-tinued Fraction expansion of z is ultimately periodic if and only if z is an irrational quadraticnumber over Q(i).

A well known result by Evariste Galois (1828) states a real number α > 1 with a purelyperiodic continued fraction expansion is a quadratic irrational whose Galois Conjugate β satisfies−1 < β < 0 and vice versa. Such quadratic irrationals are called reduced. We provide a similarresult for HCF.

Theorem 3.2. Let ξ = [a0;a1, a2, . . .] be a quadratic irrational over Q(i) and η ∈ C its Galoisconjugate over Q(i).

1. If (an)∞n=0 is purely periodic, then ∣η∣ < 1.

2. If ∣ξ∣ > 1, η ∈ F and ∣an∣ ≥ √8 for every n ∈ N0, then ξ has purely periodic expansion.

3. The conditions η ∈ F and (∀n ∈ N ∣an∣ ≥ √8) cannot be removed from the second point. In

fact, there are infinitely many pairs ξ, η such that

i. η /∈ F, ∣an∣ < √8 for some n, and (an)∞n=0 is not purely periodic,

ii. η /∈ F, ∣an∣ ≥ √8 for all n, and (an)∞n=0 is not purely periodic,

iii. η ∈ F, ∣an∣ < √8 for some n, and (an)∞n=0 is not purely periodic.

If (an)∞n=0 is a purely periodic sequence, we denote it by (an)∞n=0 = (a0, . . . , am−1). We willtacitly assume that m ∈ N is minimal with respect to the condition an = an+m for every n ∈ N.We will call a valid purely periodic sequence (a0, a1, . . . , am−1) reversible if (am−1, . . . , a1, a0)is valid. Note that every purely periodic sequence whose terms have absolute value at least

√8

is reversible.

Proof. 1. Suppose that ξ = [a0;a1, . . . , am−1]. For any j ∈ N we have

ξ = pmj−1ξ + pmj−2qmj−1ξ + qmj−2 Ô⇒ qmj−1ξ2 + (qmj−2 − pmj−1)ξ − pmj−2 = 0.

Dividing by qmj−1 we obtain a monic polynomial P ∈ Q(i)[X] satisfied by the irrational ξ,hence

ξ2 − (pmj−1qmj−1 −

qmj−2qmj−1) ξ −

pmj−2qmj−2 = 0 Ô⇒ η = pmj−1

qmj−1 − ξ −qmj−2qmj−1 .

And we conclude thatlimj→∞

qmj−2qmj−1 = −η, ∣η∣ ≤ 1. (6)

In order to show that ∣η∣ < 1 we check two cases: ∣am−1∣ ≥ 2 and ∣am−1∣ = √2.


−4. −3. −2. −1. 1. 2

−2.

−1.

1.

2.

3.

4.

0

Figure 4: Exclusion by laws of succession (red) and rationality (blue).

§First case. ∣am−1∣ ≥ 2. Assume that ∣qjm−1/qjm−2∣ < φ holds for large j. For such j,Lemma 2.1 implies that φ ≤ ∣qjm−2/qjm−3∣, so ∣qjm−3/qjm−2∣ ≤ φ−1 and

∀j ∈ N ∣qjm−1qjm−2 ∣ = ∣am−1 + qjm−3

qjm−2 ∣ ≥ 2 − 1

φ> 1,

so ∣η−1∣ > 1. When ∣qjm−1/qjm−2∣ ≥ φ > 1, we also get ∣η−1∣ ≥ φ > 1.

§Second case. ∣am−1∣ = √2. By symmetry, we may suppose that am−1 = 1+ i. The element

am−2 must exist, for (i+ 1, i+ 1) is not a valid segment. Since (∣qn∣)n≥0 is strictly increasing,

qjm−1qjm−2 = 1+ i+ qjm−3

qjm−2 ∈ E(0,1)∩D(1+ i,1) Ô⇒ qjm−2qjm−3 = am−2 + qjm−4

qjm−3 ∈ E(−1+ i,1)∩E(0,1).Assume for contradiction that ∣η∣ = 1. Then, since D(am−2,1) has to intersect the boundaryof E(1 + i,1) ∪E(0,1) and ∣am−2∣ ≥ √

2, the only possibilities for am−2 are

1 + i, 2i, −1 − i, −1 + 2i, −1 + 3i, −2, −2 + i, −2 + 2i, −3 + i. (7)

(see Figure 4.) The laws of succession of the HCF exclude the options 1 + i, 2i,− 1 − i ,−1 +2i, −2, −2 + i. The intersection D(−1 + i,1) ∩D(−1 + 3i,1) is −1 + 2i, so by taking limits, wewould have

η = 1 + i + 1−1 + 2i∈ Q(i),

which is impossible. Similarly, we can dismiss −3 + i. Hence, am−2 has to be −2 + 2i and

qjm−2qjm−3 = −2 + 2i + qjm−4

qjm−3 ∈ D(0,1) ∩E(−1 + i,1) Ô⇒ qjm−4qjm−3 ∈ D(0,1) ∩E(1 − i,1).

With a similar argument and the observations made in (4) and (5), we conclude that am−3exists and has to be equal to am−3 = 2+2i. Continuing in this way, we obtain that the periodhas to alternate between −2 + 2i and 2 + 2i. By pure periodicity, 1 + i eventually appears.However, the previous argument shows that 1+ i and 1− i are never valid options. Therefore,we can conclude ∣η∣ < 1.

2. Take the notation as in the statement. Conjugate over Q(i) the sequence (ξj)j≥0 given by

ξ0 ∶= ξ, ∀n ∈ N0 ξn+1 = 1

ξn − an ∈ F


to obtain

ζ0 ∶= η, ∀n ∈ N0 ζn+1 = 1

ζn − an .Let (kj)j≥0 be given by

k0 = 1, ∀n ∈ N0 kn+1 = √8 − 1

kn.

It is not hard to show inductively that kn →√2+1 as n→∞, that kn > 2 for n ≥ 2, and that∣ζn∣ ≤ k−1n for all n.

Assume that ξ was not purely periodic. Write

ξ = [a0;a1, . . . , an, an+1, . . . , am+n]with an ≠ an+m. Define ξ′ = [an+1, . . . , am+n] = ξn+1 and call η′ = ζn+1 its Galois conjugate.The proof of the first point tells us that we have

− 1

η′ = [am+n;am+n−1, . . . , an+1].If we had n ≥ 2, the left-most term in

ζn − ζn+m = an + 1

ζn+1 − an+m − 1

ζn+m+1 = an − an+mwould belong to D(0,1), while the right-most term would be a non-zero Gaussian integer.Therefore, we must have either n = 0 or n = 1. Suppose n = 1. Conjugate ξ1 = a1 + 1/ξ′ to getζ1 = a1 + 1/η′ and

− ζ1 = − 1

η′ − a1 = [am+1 − a1;am, . . . , a2, am+1, . . . , a2] ∈ (am+1 − a1) + F. (8)

The inequality ∣ζ1∣ ≤ k−11 = (√8 − 1)−1 along with am−1 ≠ a1 and the previous contention give∣am+1 − a1∣ = 1. Assume that am+1 − a1 = 1 (the other cases are treated similarly). Then, wehave

−ζ1 − 1 ∈ D(−1, (√8 + 1)−1) ∩ F Ô⇒ 1−ζ1 − 1∈ D((√8 − 1)2√2−8(√2 − 1) , (

√8 − 1)√2

8(√2 − 1) ) ∩ F−1.

Call c0 and ρ0, respectively, the center and the radius of the last disc. Direct calculationsgive 0.7 < ρ0 < 0.8 and −1.5 < c0 < 1.4, so

a ∈ Z[i]∗ ∶ D(c0;ρ0) ∩ F−1 ∩ (a + F) ≠ ∅ = −2 + i,−2,−2 − i.Therefore, in view of (8), we would have ∣am∣ ≤ √

5, contradicting the lower bound hypothesis.The only possibility left is n = 0, but in this case we would have ζ1 = η′ and

η = a0 + 1

ζ1= a0 − −1

η′ = a0 − am − [0;am−1, . . . , a0, am, . . . , a0] ∈ F,which implies– because the last term belongs to F– that am = a0, a contradiction. Therefore,(an)∞n=0 is purely periodic.

3. i. Take M = M1 − iM2,N = N1 − iN2 ∈ Z[i] with N1,M1 > 0 and ∣M ∣, ∣N ∣ large. Since(an)n≥1 = (M,1 + i,N,2 + 4i) is valid and reversible, we can consider the Galois conjugateirrational quadratics ξ′ and η′ with HCF expansion

ξ′ = [M ; 1 + i,N,2 + 4i], − 1

η′ = [2 + 4i;N,1 + i,M].


Define ξ = 3 + 4i + 1/ξ′ whose Galois conjugate is

η = 3 + 4i + 1

η′ = 3 + 4i − [2 + 4i;N,1 + i,M]= [1;−N,−1 − i,−M,−2 − 4i,−N,−1 − i,−M].

We have a quadratic irrational ξ, ∣ξ∣ > 1, whose HCF is not purely periodic and thathas elements with absolute value less that

√8. The Galois conjugate of ξ, η, belongs to

D(0,1) ∖ F because R(−N) < 0.

The core of the construction is to chose a reversible sequence (an)n≥1 = (a1, . . . , am) withR(am−1) < 0 and pick a0 = am + 1.

ii. The argument of the previous point can be easily adapted.

iii. The classical theory of regular continued fractions provide the first examples. Sincereduced quadratic irrationals are dense in (1,∞), they are dense in (1,2.5). Take areduced quadratic irrational α with 1 < α < 2.5. The HCF of α = [a0;a1, a2, . . .] cannot bepurely periodic, because 1 < α < 1.5 implies a0 = 1 and 1.5 < α < 2.5 implies a0 = 2, a1 < 0.However, the Galois conjugate of α in Q, and hence in Q(i), lies in (−1,0) ⊆ D. Anexplicit example is the Golden Ratio φ

φ = 1 +√5

2= [2;−3,3], −1

φ= 1 −√

5

2= [−1; 3,−3].

Although these examples do not say that√

8 is best possible, they do provide a strategyto build infinitely many numbers showing it. For any M = M1 + iM2 ∈ Z[i] ∩ F−1 withM1 > 0 define

ν ∶= ν(M) ∶= ⟨2 + i;−2 + i,M⟩The purely periodic sequence (2 + i;−2 + i,M) is not valid, because any valid prefix ofthe form (2 + i;−2 + i,M,2 + i,−2 + i,N) has RN < 0. The convergence of the continuedfraction defining ν follows from the validity of (M,−2 + i,2 + i) and the symmetry of thecontinuats Kn(a1, . . . , an) =Kn(an, . . . , a1) (See Proposition 1.1, [He06]). We claim thatthe HCF of ν is

ν = [2 + i;−2 + i,M + 1,−2 + i,2 + i,M].To verify the equality, take ξ ∶= [−2 + i; 2 + i,M]. The purely periodic expansions of ξand ν yield

((−2 + I)M + 1)ν2 + (−4 + 4M)ν + 4 = 0, ((2 + I)M + 1)ξ2 + (4 + 4M)ξ + 4 = 0.

Therefore, we can obtain a polynomial over Z[i] solved by 1 + ξ−1 and ν−1. Afterlocating both points in the complex, we can conclude that 1 + ξ−1 = ν−1. Finally, theGalois conjugate η of ν is

η = −[0;M,−2 + i,2 + i] ∈ F ⊆ D.

Remark. The Galois Conjugate η of ζ = [a0;a1, . . . , am] satisfies

η = − ⟨0;am, . . . , a1, a0⟩ ,which is not a surprise at all. This expansion, however, may fail to be a HCF. When (an)∞n=0 hassimple motives for not to be reversible, we can compute the HCF of η with a singularization2

identities like

a + 1

2 + 1

b

= a + 1 + 1

−2 + 1

b + 1

, a + 1

1 + i + 1

b

= a + (1 − i) + 1

−(1 + i) + 1

b + 1 − i.

2We borrow the term singularization from [IoKr02].

4 BOUNDED HURWITZ CONTINUED FRACTIONS 10

for any a, b ∈ C. A concrete example is given by

ζ = [5 + 6i;−3 + 2i,2,9 + 4i]and its Galois Conjugate η. The continued fraction of ζ is not reversible. Nevertheless, we canget the HCF of −η by reversing the period and applying the first singularization formula

η = −[0; 10 + 4I,−2,−2 + 2i,5 + 6i].Once we know why a sequence is not reversible, it is not hard to determine the process to obtaina valid expansion. The difficulty is in determining the source of non-reversibility.

4 Bounded Hurwitz Continued Fractions

Minkowski’s First Convex Body Theorem implies a complex version of a famous corollary toDirichlet’s Theorem on Diophantine Approixmation. Namely, a complex number ζ is irrationalif and only if there are infinitely many co-prime Gaussian integers p and q such that

∣ζ − pq∣ < 4

π

1∣q∣2 . (9)

Lester Ford showed in 1920- in a similar spirit of Hurwitz Theorem on diophantine approximation-that the constant 4/π can be replaced by (√3)−1 and that this is best possible. As in the realcase, a complex number is badly approximable if (9) cannot be improved, in some sense.

Definition 4.1. A complex number z ∈ C if badly approximable if there is a C > 0 such thatevery p, q ∈ Z[i], q ≠ 0, satisfy

∣z − pq∣ > C∣q∣2 .

The set of badly approximable complex numbers is denoted BadC.

BadC shares some properties with its real counterpart. For instance, it has Lebesgue mea-sure 0, it is 1

2 -winning in the sense of Schmidt games ([DoKr03]), and so it is of full Hausdorffdimension. We can also characterize BadC in terms of Hurwitz Continued Fractions.

Theorem 4.1. The set of badly approximable complex number is

BadC = z = [a0;a1, a2, . . .] ∈ C ∶ ∃M > 0 ∀n ∈ N0 ∣an∣ ≤M .Most of the standard argument ([Kh]) in the real version of Theorem 4.1 works in our context

too. However, we need to be sure that HCF convergents do provide good approximations.Although now convergents are not best approximations, they still have some desirable properties.We say that p/q ∈ Q(i) is a good approximation to ζ ∈ C if

∣qζ − p∣ = min∣q′x − p′∣ ∶ q′, p′ ∈ Z[i] ∣q′∣ ≤ ∣q∣ .We say that p/q ∈ Q(i) is a best approximation to ζ ∈ C if

∀p′, q′ ∈ C ∣q′∣ < ∣q∣ Ô⇒ ∣qζ − p∣ < ∣q′ζ − p′∣.The next result is Theorem 1 of [La73].

Theorem 4.2 (Lakein, 1973). Let ζ be a complex number. Every convergent of ζ is a goodapproximation to ζ. Moreover, for almost every ζ ∈ C (with respect to the Lebesgue measure)every convergent is a best approximation to ζ.

4 BOUNDED HURWITZ CONTINUED FRACTIONS 11

Proposition 4.1. Let ξ = [a0;a1, a2, . . .] be an irrational complex number and (pn)∞n=0, (qn)∞n=0its Q-pair. Then, for some absolute constant we have

∀n ∈ N 1(∣zn+1∣ + 1) ∣qn∣2 < ∣ξ − pnqn

∣ ≪ 1∣qnqn+1∣ . (10)

Proof of Proposition 4.1. An inductive argument (Proposition 1, [DaNo14]) gives us that qnzn+1+qn−1 = z(pnzn+1 + pn−1) for every n ∈ N, therefore

z = pnzn+1 + pn−1qnzn+1 + qn−1 = pn

qn+ (−1)nq2n (zn+1 + qn−1

qn) = pn

qn+ (−1)nq2n (an+1 + 1

zn+2 + qn−1qn

)The left inequality in (10) follows immediately from the strict monotonicity of (∣qn∣)n≥0. Forthe right inequality, we use zn+1 = an+1 + z−1n+2 to obtain

z − pnqn

= (−1)nqn(qnzn+1 + qn−1) = (−1)n

qn (qn+1 + qn 1zn+2 ) = (−1)n

qnqn+1 (1 + qnqn+1

1zn+2 ) . (11)

Since (∣qn∣)n≥0 is strictly increasing and z−1n+2 ∈ F ⊆ D(0,2− 12 ), we can uniformly bound by below∣1 + qn

qn+11

zn+2 ∣ as follows

∣1 + qnqn+1

1

zn+2 ∣ ≥ 1 − √2

2> 0.

The previous inequality plugged into (11) yields the result.

Proof of Theorem 4.1. We start by showing ⊇. Let z = [a0;a1, a2, . . .] be a complex numberwhose elements are bounded by M > 0. Then,

∀n ∈ N ∣z∣ = ∣an + 1

ζn+1 ∣ ≤M + 1

and, by the left inequality of Lemma 10, we have for c1 = (M + 2)−1 that

∀n ∈ N c1∣qn∣2 ≤ ∣z − pnqn

∣ .Now, take p/q ∈ Q(i) in lowest terms and n ∈ N such that ∣qn−1∣ < ∣q∣ ≤ ∣qn∣. By Theorem 4,∣qnz − pn∣ ≤ ∣qz − p∣, so our choice of n gives

∣z − pq∣ ≥ ∣qn∣∣q∣ ∣z − pn

qn∣ ≥ c1∣qn∣2 > c1∣q∣2 ∣qn−1∣2∣qn∣2 = c1∣q∣2 1

∣an + qn−2qn−1 ∣2

≥ c1∣q∣2(M + 1)2 .Therefore, z ∈ BadC satisfies with C = (M + 1)−1(M + 2)−2.

In order to show the other contention, take z ∈ BadC and let C = C(z) > 0 be the constantfrom the definition. By Lemma 10, for some constant c > 0 and any n ∈ N we have

C∣qn∣2 ≤ c∣qnqn+1∣ Ô⇒ ∣an+1 + qn−1qn

∣ = ∣qn+1qn

∣ ≪ 1 Ô⇒ ∣an+1∣ ≪ 1.

The last implication follows from ∣qn−1∣ < ∣qn∣. The constants in ≪ depend on z.

This characterization of BadC allows us to restate Theorem 1.3 as follows.

Theorem 4.3. There are complex, badly approximable, algebraic numbers of arbitrary evendegree over Q(i).

Recently, Robert Hines ([Hi17]) gave a different proof of Theorem 4.1. His argument alsorelies on R. Lakein’s work but avoids Lemma 10.

5 CONSTRUCTION OF TRANSCENDENTAL COMPLEX NUMBERS 12

5 Construction of Transcendental Complex Numbers

Let us recall that the length of a finite word x is the number of terms it comprises and it isdenoted by ∣x∣. In general, let A ≠ ∅ be a finite alphabet and a an infinite word over A. Asnoted in [BuKi15], rep(a) < +∞ is equivalent to the existence of three sequences of finite wordsin A, (Wn)n≥1, (Un)n≥1, and (Vn)n≥1, such that

i. For every n the word WnUnVnUn is a prefix of a,

ii. The sequence (∣Wn∣ + ∣Vn∣)/∣Un∣ is bounded above,

iii. The sequence (∣Un∣)n≥1 is strictly increasing.

The behaviour of (Wn)n≥1 prompts two different cases:

lim infn→∞ ∣Wn∣ < +∞ ∨ lim inf

n→∞ ∣Wn∣ = +∞.Our main result is the following theorem.

Theorem 5.1. Let a = (aj)j≥1 be a non-periodic valid sequence such that

repa < +∞.Call ζ = [0;a1, a2, a3, . . .] and let (Wn)n≥0, (Un)n≥0, (Vn)n≥0 be as above.

i. If lim infn→∞ ∣Wn∣ < +∞, then ζ is transcendental.

ii. If lim infn→∞ ∣Wn∣ = +∞ and ∣an∣ ≥ √

8 for every n ∈ N, then ζ is transcendental.

A drawback of our analogue of Theorem 1.1 is that the slight weakening leads us to a lotless elegant statement.

5.1 Preliminary Lemmas

Keep the notation of Theorem 5.1 and call (pn)∞n=0, (qn)∞n=0 the Q-pair of ζ. For simplicity’ssake, consider the sequences of non-negative integers (wn)n≥1, (un)n≥1, and (vn)n≥1 given by

∀n ∈ N wn = ∣Wn∣, un = ∣Un∣, vn = ∣Vn∣.We can control the growth of (qn)n≥0 throught the combinatorial conditions on (Wn)n≥1,(Un)n≥1, and (Vn)n≥1.Lemma 5.2. There exists ε > 0 such that for every n ∈ N

ψun ≥ ∣qwnqwn+un+vn ∣ε,with ψ ∶= (1+√5

2 ) 13.

Proof. The boundedness of (aj)j≥1 and ((vn +wn)/un)n≥0 allows us to define

M ∶= 1 + lim supn→∞ ∣qn∣ 1n , N ∶= 2 + lim sup

n→∞2wn + vn

un.

By Lemma 2.1,

ψ = (MwnMwn+un+vn) logM ψ

2wn+un+vn > ∣qwnqun+vn+wn ∣ logM ψ

2wn+un+vn ,

and ε = logψN logM works:

ψun > (∣qwn ∣ un2wn+un+vn ∣qun+vn+wn ∣ un

2wn+un+vn )logM ψ > ∣qwnqun+vn+wn ∣ logψN logM .


As in [Bu15], our main tool is an adequate version of Schmidt’s Subspace Theorem. In[Sch76], Wolfgang Schmidt obtained his Subspace Theorem for number fields, but we will con-sider just consider a special case.

Theorem 5.3 (Schmidt’s Subspace Theorem for Number Fields). Let L1, . . . ,Lm and M1, . . . ,Mm

be two sets of m linearly independent linear forms in m variables with algebraic coefficients.For any ε > 0 there are proper subspaces of Q(i)m, T1, . . . , Tk, such that for every β =(b1, . . . , bk) ∈ Z[i] with β ≠ 0

RRRRRRRRRRRm∏j=1Lj(β)RRRRRRRRRRR

RRRRRRRRRRRm∏j=1Mj(β)RRRRRRRRRRR ≤

1∥β∥ε∞ Ô⇒ β ∈ k⋃j=1Tj ,

where ∥β∥∞ = max∣b1∣, . . . , ∣bk∣, and β = (b1, . . . , bm).

5.2 Proof of Theorem 5.1

We can restate the two cases in a much simpler way after taking, if necessary, appropriatesub-sequences. Instead of lim infnwn < +∞ we will think of (wn)n≥1 as constant. And whenlim infnwn < +∞, we will suppose that (wn)n≥1 is strictly increasing and that awn ≠ awn+un+vn .We do not lose generality with the later restriction. Indeed, if we had awn = awn+un+vn = a, wecould take finite words (possibly empty) W ′, V ′ such that W = W ′a and UV = UV ′a. Hence,WnUnVnU =W ′aUnV ′aUn = W U V U where W =W ′, U = aUn, and V = V ′.

Throughout the proof, the constants implied by ≪ will depend on ζ.

§.(wn)n≥1 is constant. Write k = w1 ∈ N0 and W =W1. We can assume that k = 0, becausethe transcendence of any element in [an;an+1, an+2, . . .] ∶ n ∈ N implies the transcendence ofall the others.

i. Write sn = un + vn and define the sequence of irrational quadratic numbers (ζ(n))n≥1 by

∀n ∈ N ζ(n) = [0; b1, b2, b3, . . .] = [0;UnVnUnVnUnVn . . .],where the second term means

∀j ∈ N bj = (UnVn)r, 0 ≤ r ≤ sn − 1, r ≡ j (mod sn).The validity of the sequences defining ζ(n) is not quite evident; in fact, some of them mightnot be. Nevertheless, after taking a subsequence, we can assume that they are valid. Wegive a broad sketch of the argument and leave the details to the reader. In general, supposethat for X = x1 . . . xm, Y = y1 . . . yn the word XYX is a valid prefix but 0XYXY is not.Without loss of generality, we can assume that 0XYXy1 is not a valid prefix. Since xmy1appears once but it is forbidden the second time, ∣xm∣ ∈ √5,

√8. Therefore, the maximal

feasible region for a residue following xm either has the form F2 or F3 (see Figure 2). Afterverifying separately each case for ∣xm∣, we conclude that the first m − 1 terms of X mustalternate between ik(2 + 2i) and ik(−2 + 2i) for some fixed k ∈ 1,2,3,4. Moreover, wemust have ∣yj ∣ = √

5 for some j. This observation, un → ∞, and lim supn→∞ vn/un < +∞imply that the sequences 0UnVn are valid for sufficiently large n.

The first un + sn terms of the HCF of ζ and ζ(n) coincide, then (cfr. Proposition 4.1)

∣ζ − ζ(n)∣ < 1∣qun+sn ∣2 .As in the real case, each ζ(n) satisfies the polynomial

Pn(X) ∶= qsn−1X2 + (qsn − psn−1)X − psn .


Therefore, we can bound Pn(ζ) as follows

∣Pn(ζ)∣ = ∣Pn(ζ) − Pn(ζ(n))∣= ∣qsn (ζ − ζ(n)) (ζ + ζ(n)) + (qsn − psn−1) (ζ − ζ(n))∣≪ ∣qsn ∣ ∣ζ − ζ(n)∣ + 2∣qsn ∣ ∣ζ − ζ(n)∣ ≪ ∣qsn ∣∣qun+sn ∣2 .

ii. First application of the Subspace Theorem. Consider the sequence of points in Z[i]4,(xn)∞n=1, given by

∀n ∈ N xn = (qsn−1,−psn−1, qsn ,−psn), ∥xn∥∞ = ∣qsn ∣.Define two collections of independent linear forms in C, L 1 = L 1

1 ,L12 ,L

13 ,L

14 and

M 1 = M 11 ,M

12 ,M

13 ,M

14 in the variables X = (X1,X2,X3,X4) and X = (X1, X2, X3, X4),

respectively, by

L 11 (X) = ζ2X1 − ζ(X2 +X3) +X4, M 1

1 (X) = ζ2X1 − ζ(X2 + X3) + X4,

L 12 (X) = ζX1 −X2, M 1

2 (X) = ζX1 − X2,

L 13 (X) = ζX1 −X3, M 1

3 (X) = ζX1 − X3,

L 14 (X) =X1, M 1

4 (X) = X1. (12)

Evaluating the product of the forms L 1 in xn we have that

∣L 11 L 1

2 L 13 L 1

4 (xn)∣ = ∣Pn(ζ)∣∣ζqsn−1 − psn−1∣∣ζqsn−1 − qsn ∣∣qsn−1∣≪ ∣qsn ∣2∣qsn+un ∣2 < 1

ψ2un.

Hence, by Lemma 5.2, there exists ε > 0 such that

∀n ∈ N ∣L 11 L 1

2 L 13 L 1

4 (xn)∣ ≪ 1∥xn∥ε∞ . (13)

By construction, ∣M 1j (xn)∣ = ∣L 1

j (xn)∣ for all n and j ∈ 1, . . . ,4, where the bar denotescomplex conjugation component-wise. From Theorem 5.3 we conclude the existence of avector 0 ≠ x = (x1, x2, x3, x4) ∈ Z[i]4 and an infinite set N1 ⊆ N such that

∀n ∈ N1 x1qsn−1 + x2psn−1 + x3qsn + x4psn = 0.

Dividing the last expression by qsn−1 we get

∀n ∈ N1 x1 + x2 psn−1qsn−1 + x3

qsnqsn−1 + x4

psnqsn

qsnqsn−1 = 0.

Since the Gaussian integers x2 and x4 cannot both be 0 (ζ is irrationals), we can define

ξ = limn→∞n∈N1

qsnqsn−1 = −ζx2 + x1

ζx4 + x3 .The irrationality of ξ follows after obtaining infinitely many complex rational numbersapproaching to ξ at a certain rate. For every n ∈ N1 we have that

∣ξ − qsn−1qsn

∣ = RRRRRRRRRRRRx1 + ζx3x2 + ζx4 −

x1 + psnqsn

x3

x2 + psn−1qsn−1 x4

RRRRRRRRRRRR≤ ∣x1∣ RRRRRRRRRRRR

1

x1 + ζx4 − 1

x2 + x4 psn−1qsn−1

RRRRRRRRRRRR + ∣x3∣ RRRRRRRRRRRRζ

x2 + x1ζ −psnqs−n

x2 + psn−1qsn−1 x4

RRRRRRRRRRRR≪ ∣ζ − psn

qsn∣ + ∣ζ − psn−1

qsn−1 ∣ + ∣psn−1qsn−1 −

psnqsn

∣ ≪ 1∣qsnqsn−1∣ . (14)


If ξ were rational, ξ = a/b for some co-prime a, b ∈ Z[i], we would have

1∣bqvn−1 ∣ ≤ ∣ξ − qvn−1qvn

∣ ≪ 1∣qvn−1qvn ∣ Ô⇒ ∣qvn−1∣ ≪ ∣b∣,for qvn−1 and qvn co-prime. In other words, (∣qvn−1∣)n≥0 would be bounded, which is absurd.Hence, ξ ∈ Q(i, ζ) is irrational.

iii. Second Application of the Subspace Theorem. Consider the linear forms L 2 = L 21 ,L

22 ,L

23

and M 1 = M 21 ,M

22 ,M

23 in the variables Y = (Y1, Y2, Y3) and Y = (Y1, Y2, Y3) given by

L 21 (Y) = ξY1 − Y2, M 2

1 (Y) = ξY1 − Y2,L 2

2 (Y) = ζY1 − Y3, M 22 (Y) = ζY1 − Y3,

L 23 (Y) = Y2, M 2

3 (Y) = Y2.Define sequence (yn)n≥0 in Z[i]3 by

∀n ∈ N yn ∶= (qun+vn , qun+vn−1, pun+vn), ∥yn∥∞ = ∣qun+vn ∣.Since

∣M 21 M 2

2 M 23 (yn)∣ = ∣L 2

1 L 22 L 2

3 (yn)∣= ∣(qun+vnξ − qun+vn−1)(qun+vnζ − pun+vn)qun+vn−1∣≤ ∣qun+vn−1∣∣qun+vnqun+vn−1∣ =

1∥yn∥∞ ,by Lemma 5.2, there is an ε > 0 such that

∀n ∈ N1

RRRRRRRRRRR3∏j=1L 2

j (yn)RRRRRRRRRRR ∣3∏k=1M 2

k (yn)∣ ≤ 1∥yn∥ε .Schmidt’s Subspace Theorem yields the existence of a non-zero (y1, y2, y3) ∈ Z[i]3 and aninfinite set N2 ⊆ N1 such that

∀n ∈ N2 qvny1 + qvn−1y2 + pvny3 = 0.

Dividing by qvn and taking limits along N2 we obtain

y1 + ξy2 + ζy3 = 0, (15)

and, since ζ is irrational, y2y3 ≠ 0.

iv. Third application of the Subspace Theorem. Define L 3 = L 31 ,L

32 L 3

3 and M 3 = M 31 ,M

32 M 3

3 in the variables Z = (Z1, Z2, Z3) and Z = (Z1, Z2, Z3) by

L 31 (Z) = ξZ1 −Z2, M 3

1 (Z) = ξZ1 − Z2,

L 32 (Z) = ζZ2 −Z3, M 3

2 (Z) = ζZ2 − Z3,

L 33 (Z) = Z2, M 3

3 (Z) = Z2,

and the sequence (zn)n≥1 by

∀n ∈ N zn = (qsn , qsn−1, psn), ∥zn∥∞ = ∣qsn ∣.We get from (14) and the upper bound in (10) that

∣M 31 M 3

2 M 33 (zn)∣ = ∣L 3

1 L 32 L 3

3 (zn)∣= ∣(qsnξ − qsn−1)(ζqsn−1 − psn−1)qsn−1∣≪ ∣qsn−1∣∣qsn−1qsn ∣ =

1∥zn∥∞ .


Once again, by Schmidt’s Subspace Theorem, there is a non-zero triplet (z1, z2, z3) ∈ Z[i]3and N3 ⊆ N2 infinite for which

∀n ∈ N3 z1qsn + z2qsn−1 + z3psn = 0.

Dividing by qsn−1 and taking the limit when n→∞ along N3, we get

1

ξz1 + z2 + ζz3 = 0. (16)

Since ζ and ξ are irrational, z1z3 ≠ 0. Combining (15) and (16) we get

z1y2 = (z2 + ζz3)(y1 + ζy3),with z3y3 = 0. This contradicts [Q(i, ζ) ∶ Q(i)] ≥ 3. Therefore, ζ is transcendental.

§.(wn)n≥1 is strictly increasing. As before, we start by approximating ζ with quadraticirrationals. However, this time they will not have purely periodic expansions.

i. Define the sequence (ζ(n))n≥1 by

∀n ∈ N ζ(n) = [0;WnUnVnUnVnUnVn . . .].Tedious but straight forward computations show that each ζ(j) satisfies the polynomial

Pn(X) ∶= ∣qwn−1 qwn+un+vn−1qwn qwn+un+vn ∣X2 − (∣qwn−1 pwn+un+vn−1

qwn pwn+un+vn ∣ + ∣pwn−1 qwn+un+vn−1pwn qwn+un+vn ∣)X+

+ ∣pwn−1 pwn+un+vn−1pwn pwn+un+vn ∣

The first wn + 2un + vn elements of ζ and ζ(n) coincide, so

∀n ∈ N ∣ζ − ζ(n)∣ ≪ 1∣qwn+2un+vn ∣2 .The previous equality, the strict increase of (∣qk∣)k≥0, and ∣pk∣ < ∣qk∣ for any k (because ζ ∈ F)imply that

∀n ∈ N ∣Pn(ζ)∣ = ∣Pn(ζ) − Pn (ζ(n))∣ ≪ ∣qwn+un+vn ∣∣qwnq2wn+2un+vn ∣ . (17)

ii. First application of the Subspace Theorem. Define the forms L 1 and M 1 as in (12). Letxn = (xn,1, xn,2, xn,3, xn,4) in Z[i]4 be the sequence

xn,1 ∶= ∣qwn−1 qwn+un+vn−1qwn qwn+un+vn ∣ , xn,2 ∶= ∣qwn−1 pwn+un+vn−1

qwn pwn+un+vn ∣ ,xn,3 ∶= ∣pwn−1 qwn+un+vn−1

pwn qwn+un+vn ∣ , xn,4 ∶= ∣pwn−1 pwn+un+vn−1pwn pwn+un+vn ∣ .

It can be easily proven that

∀n ∈ N ∥xn∥∞ ≪ ∣qwnqwn+un+vn ∣. (18)

For example, since (∣qj ∣)j≥0 is strictly increasing and ∣pj ∣ ≤ ∣qj ∣ for every j, we have

∣xn,2∣ = ∣qwn−1pwn+un+vn − pwn+un+vn−1qwn ∣ ≤ ∣qwn ∣ (∣pwn+un+vn ∣ + ∣pwn+un+vn−1∣)≪ ∣qwnqwn+un+vn ∣.


Let ε > 0 be as in Lemma 5.2. For every xn

4∏j=1 ∣M 1

j (xn)∣ = 4∏j=1 ∣L 1

j (xn)∣ = ∣Pn(ζ)∣ ∣ζxn,1 − xn,2∣ ∣ζxn,1 − xn,3∣ ∣xn,1∣≪ ∣ qwn+un+vn

qwnq2wn+2un+vn

qwnqwn+un+vn

qwn+un+vnqwn

qwnqwn+un+vn∣= ∣ q2wn+un+vnq2wn+2un+vn ∣ ≪

1∣qwnqwn+un+vn ∣ε =1∥xn∥ε .

Therefore, there exist 0 ≠ x = (x1, x2, x3, x4) ∈ Z[i]4 and an infinite set N ′1 ⊆ N such that

∀n ∈ N ′1 0 =x1 ∣qwn−1 qwn+un+vn−1

qwn qwn+un+vn ∣ + x2 ∣qwn−1 pwn+un+vn−1qwn pwn+un+vn ∣

+ x3 ∣pwn−1 qwn+un+vn−1pwn qwn+un+vn ∣ + x4 ∣pwn−1 pwn+un+vn−1

pwn pwn+un+vn ∣ . (19)

Let (Qn)n≥1 and (Rn)n≥1 be

∀n ∈ N Qn ∶= qwn−1qwn+un+vnqwnqwn+un+vn−1 , Rn ∶= ζ − pn

qn.

Divide (19) by qwnqwn+un+vn−1 to obtain for every n ∈ N1 the equation

0 = x1(Qn − 1) + x2 (Qn(ζ −Rwn+un+vn) − (ζ −Rwn+un+vn−1))++ x3 (Qn(ζ −Rwn−1) − (ζ −Rwn))++ x4 (Qn(ζ −Rwn−1)(ζ −Rwn+un+vn) − (ζ −Rwn)(ζ −Rwn+un+vn−1)) . (20)

From (20) we obtain

0 = (Qn − 1)(x1 + (x2 + x3)ζ + ζx4) + η(n),where η(n)→ 0 when n→∞ along N ′

1, because

η(n) = x2((Qn − 1)Rwn+un+vn +Rwn+un+vn−1 −Rwn+un+vn)++ x3((−Qn + 1)Rwn−1 +Rwn −Rwn−1)++ x4(Qn − 1)(−(Rwn−1 +Rwn+un+vn) −Rwn−1Rwn+un+vn)++ (ζ −Rwn−1)(ζ −Rwn+un+vn) − (ζ −Rwn)(ζ −Rwn+un+vn−1).We conclude that

limn→∞(Qn − 1) (x1 + (x2 + x3)ζ + ζ2x4) = 0

The boundedness of (an)∞n=0 allows us to assume the existence of the following limits

α = limn→∞n∈N

qwnqwn−1 , β = lim

n→∞n∈N

qwn+un+vnqwn+un+vn−1 , (21)

and that a = awn ≠ awn+un+vn = b are constant. We can assure that (cfr. (1) and Proposition2.1)

qwnqwn−1 = [a;awn−1, . . . , a0] ∈ a +K,

qwn+un+vnqwn+un+vn−1 = [b;awn+un+vn−1, . . . , a0] ∈ b +K,

and– because a +K and b +K are disjoint compact sets– α ≠ β and Qn /→ 1 as n→∞ alongN ′2. As a consequence, we get

(x1 + (x2 + x3)ζ + ζ2x4) = 0. (22)

6 FURTHER RESULTS 18

Since ζ is neither quadratic nor rational, we must have that x1 = x4 = 0, x2 = −x3, and thepolynomials Pn become

Pn(X) = ∣qwn−1 qwn+un+vn−1qwn qwn+un+vn ∣X2 − 2 ∣qwn−1 pwn+un+vn−1

qwn pwn+un+vn ∣X + ∣pwn−1 pwn+un+vn−1pwn pwn+un+vn ∣ .

Consider the sets of linear forms L 4 = L 41 ,L

42 ,L

43 and M 4 = M 4

1 ,M42 ,M

43 in the

variables Y = (Y1, Y2, Y3) and Y = (Y1, Y2, Y3) given by

L 41 (Y) = ζ2Y1 − 2ζY2 + Y3, M 4

1 (Y) = ζ2Y1 − 2ζY2 + Y3L 4

1 (Y) = ζY1 − Y2, M 41 (Y) = ζ2Y1 − Y2

L 41 (Y) = Y1, M 4

1 (Y) = Y1For n ∈ N ′

1 define

vn ∶= (∣qwn−1 qwn+un+vn−1qwn qwn+un+vn ∣ , ∣qwn−1 pwn+un+vn−1

qwn pwn+un+vn ∣ , ∣pwn−1 pwn+un+vn−1pwn pwn+un+vn ∣) .

Using the new expression for Pn and an idea similar to the one proving (17), we obtain thatfor some ε > 0RRRRRRRRRRR

3∏j=1M 4

j (vn)RRRRRRRRRRR =RRRRRRRRRRR3∏j=1L 4

j (vn)RRRRRRRRRRR= ∣Pn(ζ)(ζvn,1 − vn,2)vn,1∣ ≪ ∣qwnqwn+un+vm ∣∣q2wn+2un+vn ∣ ≪ 1∣qwnqwn+un+vn ∣ε .

Schmidt’s Subspace Theorem guarantees the existence of 0 ≠ (t1, t2, t3) ∈ Z[i]3 such thatfor some infinite N ′

2 ⊆ N ′1

t1 ∣qwn−1 qwn+un+vn−1qwn qwn+un+vn ∣ + t2 ∣qwn−1 pwn+un+vn−1

qwn pwn+un+vn ∣ + t3 ∣pwn−1 pwn+un+vn−1pwn pwn+un+vn ∣ = 0.

Dividing by qwnqwn+un+vn−1, we get

t1(Qn − 1) + t2 (Qn pwn+un+vnqwn+un+vn −

pwn+un+vn−1qwn+un+vn−1 )+

+ t3 (Qn pwn−1qwn−1

pwn+un+vnqwn+un+vn −

pwnqwn

pwn+un+vn−1qwn+un+vn−1 ) = 0.

Hence, since Qn /→ 1 when n→∞ along N ′2,

t3ζ2 + t2ζ + t1 = 0.

The previous equation contradicts [Q(i, ζ) ∶ Q(i)] ≥ 3. Therefore, ζ is transcendental.

6 Further results

Other transcendence results can be translated into our context with the pertinent modifications.An example is Theorem 6.1 (cfr. [Bu15], Theorem 11.2).

In general, we say that an infinite word a ∈ A satisfies condition (♣) if there are sequences(Wn)n≥1, (Un)n≥1, (Vn)n≥1, (Un)n≥1 of finite words in A such that

1. WnUnVnUn is a prefix of a for every n ∈ N, where Un is the word obtained by reversingUn,

2. ((∣Wn∣ + ∣Vn∣)/∣Un∣)n≥1 is bounded,

7 FINAL COMMENTS 19

3. (∣Un∣)n≥1 tends to infinity when n does.

Theorem 6.1. Let a ∈ ΩHCF satisfy Condition (♣) and ∣an∣ ≥ √8 for every n. If a is not

periodic, then ζ = [0;a1, a2, . . .] is transcendental.

Remark. i. The proof of Theorem 6.1 needs an inequality involving continuants (the denom-inators of the convergents, denoted Kn). For any valid segments a, b of length n and mrespectively such that ab is a valid prefix, we have for some absolute constant

∣Kn+m(ab)∣ ≪ ∣Kn(a)Kn(b)∣.The inequality follows from well known continued fraction identity. If x = (x1, . . . , xn) andy = (y1, . . . , ym) are two vectors of variables, then

Kn+m(xy) =Kn(x)Km(y) (1 + ⟨0;xn, . . . x1⟩ ⟨0; y1, . . . ym⟩) .For a proof, see Proposition 1.1. in [He06].

ii. Getting rid of the condition ∣an∣ ≥ √8 for every n in Theorem 6.1 poses essentially the same

problem as the corresponding bound in Theorem 5.1.

7 Final comments

Our argument for Qn /→ 1 along N ′ in the second case of Theorem 5.1 is simpler than theoriginal one. Unfortunately, it does not help to omit the lower bound and get a full analogueof Theorem 1.1. We can understand why with the help of Figure 3. Unlike regular continuedfractions, for every M > √

8 we can find arbitrarily close complex numbers z = [a0;a1, . . .] andw = [b0; b1, . . .] such that a0 ≠ b0 and sup ∣an∣, sup ∣bn∣ <M .

Acknowledgements.

Thanks to Yann Bugeaud for his invaluable comments. This research was supported by CONA-CYT (grant: 410695) and Aarhus University (St.: 2012, Pr.: 25763, Ak.: 26248).

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[ASch] A. Schmidt, Diophantine approximation of omplex numbers. Acta Math. 134 (1975),1-85

[Sch76] W.M. Schmidt, Simultaneous approximation to algebraic numbers by elements of a num-ber field. Monatshefte fr Mathematik (1975) 79: 55. https://doi.org/10.1007/BF01533775

[Sch96] W.M. Schmidt, (1996), Diophantine Approximation. Series: Lecture Notes in Mathe-matics 785. Berlin: Springer Verlag. DOI. 10.1007/978-3-540-38645-2

Gerardo Gonzalez RobertDepartment of MathematicsAarhus UniversityNy Munkegade 118, Aarhus C 8000, [email protected]

Good’s Theorem for Hurwitz Continued Fractions


Abstract

Good’s Theorem for regular continued fraction states that the set of real numbers[a0;a1, a2, . . .] such that limn→∞an = ∞ has Hausdorff dimension 1

2. We show an

analogous result for the complex plane and Hurwitz Continued Fractions. Theset of complex numbers whose Hurwitz Continued fraction [a0;a1, a2, . . .] satisfieslimn→∞ ∣an∣ =∞ has Hausdorff dimension 1, half of the ambient space’s dimension.

1 Introduction

A basic result on regular continued fractions states that every irrational number α admitsinfinitely many rational approximations p/q, p, q ∈ Z and q ≥ 1, for which

∣α − pq∣ < 1

q2.

For some real numbers the previous inequality cannot be improved in terms of theexponent of q. These numbers are called badly approximable numbers and the set ofbadly approximable real numbers is denoted by BadR. To be more precise, α ∈ R∖Q isbadly approximable if there exists c > 0 such that1

∀(p, q) ∈ Z ×N ∣α − pq∣ ≥ c

q2.

It is well known that BadR is precisely the set of irrationals whose regular continuedfraction is given by a bounded sequence. The characterization in terms of continued frac-tion allows us to show in a simple way that m1BadR = 0, where m1 denotes the Lebesguemeasure in R. However small in terms of Lebesgue measure, V. Jarnık published in 1928(Satz 4, [Ja28]) an influential result saying that BadR is rather large.

Theorem 1.1 (V. Jarnık,1928). The set BadR has full Hausdorff dimension; that is

dimH [0;a1, a2, . . .] ∈ R ∶ lim supn→∞ an < +∞ = 1.

A result related to Jarnık’s theorem was published by I.J. Good in 1941 (Theorem1, [Goo41]).

1N = n ∈ Z ∶ n ≥ 1 is the set of natural numbers and N0 ∶= N ∪ 0.1

1 INTRODUCTION 2

Theorem 1.2 (I. G. Good, 1941). The following equality holds

dimH [0;a1, a2, . . .] ∈ R ∶ limn→∞an = +∞ = 1

2.

Extensions of continued fractions and of Jarnık’s Theorem to other contexts havebeen successfully carried out (v. gr. [KTV06] or [Sc] and the references therein). Re-cently, important work has been done in the complex plane using a complex continuedfraction algorithm suggested by Adolf Hurwitz in 1887 ([Hu87]) ([DaNo14]).

The First Minkowski’s Theorem on Convex Bodies (or Dirichlet’s Pigeonhole Prin-ciple) tells us that for some absolute constant C > 0 and any complex irrational ζ (i.e.ζ ∈ C ∖Q(i)) there are infinitely many pairs (p, q) ∈ Z[i] ×Z[i]∗ such that

∣ζ − pq∣ ≤ C∣q∣2 .

As in the real case, we say that a complex irrational ζ is badly approximable if forsome c > 0 we have ∀(p, q) ∈ Z[i] ×Z[i]∗ ∣ζ − p

q∣ ≥ c∣q∣2 .

We denote by BadC the set of badly approximable complex numbers. Some propertiesof BadC are well known. For example, if m2 denotes the Lebesgue measure on C, thenm2BadC = 0 and BadC is 1

2 -winning in the sense of Schmidt games and so, it is of fullHausdorff dimension (Theorem 5.2, [DoKr03]). BadC can also be characterized in termsof Hurwitz Continued Fractions: a complex irrational ζ belongs to BadC if and onlyif its Hurwitz Continued fraction is bounded (Theorem 4.1, [Gon18]). Our main resultestablishes another similarity between Hurwitz and regular continued fractions.

Theorem 1.3. For any z ∈ C ∖Q(i) denote its Hurwitz Continued Fraction expansionby z = [a0;a1, a2, . . .], then

dimH z = [a0;a1, a2, a3, . . .] ∈ C ∶ limn→∞ ∣an∣ = +∞ = 1.

Although similarities between real and complex continued fractions abound, somedifferences have to be considered. While regular continued fractions establish an home-omorphism between the irrationals in [0,1] and NN the space of sequences associated tothe HCF process is much more complicated. Since the difficulty arises from sequences(an)n≥1 with minn≥1 ∣an∣ ≤ √

8, it is natural to ask how large are the subsets of BadCwhere a uniform lower bound is imposed on the absolute value of the terms of the HCFexpansion. In this direction, the proof of Theorem 1.3 gives us the next corollary (SeeSection 2 for the definition of F).

Corollary 1.4. For any L ∈ N with L ≥ √8 define

EL ∶= z = [0;a1, a2, a3, . . .] ∈ F ∶ ∀n ∈ N L ≤ ∣an∣ ,then

dimH (EL ∩BadC) 1 as L→∞.

2 GENERAL PROPERTIES OF HURWITZ CONTINUED FRACTIONS 3

The organization of the text is as follows. In Section 2, we define precisely theHurwitz Continued Fraction algorithm and some of its properties. In Section 3, wegive two lemmas for estimating the Hausdorff dimension of a class of Cantor sets. InSection 4 we state some preliminary lemmas concerning the Hausdorff dimension of setsobtained by imposing restrictions on the Hurwitz Continued Fraction expansions. InSection 5 we show Theorem 1.3 and Corollary 1.4. Finally, in Section 6 we prove thepreliminary lemmas of Section 4.

Notation.

i. N is the set of natural numbers, considered as the set of positive integers,

ii. if (xn)n≥1 and (yn)n≥1 are two sequences of non-negative numbers and there existssome constant C > 0 such that xn ≤ Cyn for every large n, we write xn ≪ yn. Byxn ≍ yn we mean xn ≪ yn ≪ xn. Whenever the implied constants depend on aparameter ε, say, we write xn ≪ε yn and similarly for ≍.

iii. if (X,d) is a metric space and A ⊆X, the diameter of A is ∣A∣ ∶= supd(x, y) ∶ x, y ∈A.

iv. for any complex number z we write D(z) = w ∈ C ∶ ∣z −w∣ < 1, D(z) is the closureof D(z) and C(z) is its boundary.

v. let A ⊆ C. A is the interior of A and A−1 = z−1 ∶ z ∈ A.

2 General Properties of Hurwitz Continued Fractions

Let [⋅] ∶ C → Z[i] be the function that assigns to each complex number the nearestGaussian integer (choosing the one with greater real and imaginary parts to break ties).Denote by F the inverse image of 0 under [⋅],

F ∶= z ∈ C ∶ −1

2≤Rz,Iz < 1

2 . (1)

Let T ∶ F∗ → F∗, with F∗ ∶= F ∖ 0, be given by

∀z ∈ F∗ T (z) = 1

z− [1

z] .

For any z ∈ F∗ define– as long as the operations make sense– the sequences (an)n≥0,(zn)n≥1 by

z1 ∶= z, ∀n ∈ N zn+1 = T (zn),a0 ∶= 0, ∀n ∈ N an = [z−1n ] .

The Hurwitz Continued Fraction (HCF) of z is the sequence (an)n≥1. We caneasily extend the definition to an arbitrary complex number w directly. The Hurwitz


Continued Fraction of w ∈ C is the sequence (an)n≥0 where a0 = [w] and (an)n≥1 is theHCF of w − a0. We refer to the numbers an as elements. Following [DaNo14], we callthe sequences (pn)n≥0, (qn)n≥0 given by

(p−1 p0q−1 q0

) = (1 00 1

) , ∀n ∈ N (pnqn

) = (pn−1 pn−2qn−1 qn−2)(an

1) .

the Q-pair of z.We summarize some properties of HCF in the next proposition.

Proposition 2.1. Let z ≠ 0 belong to F and denote by (an)n≥0, (pn)n≥0, (qn)n≥0 be itsassociated sequences.

i. For every n ∈ Npnqn

= [0;a1, a2, . . . , an].ii. The sequence (an)n≥0 is infinite if and only if z ∈ F ∖Q(i). In this case, we have

thatlimn→∞[0;a1, a2, . . . , an] = z.

As a consequence, HCF give an injection from F ∖Q(i) into Z[i]N.

iii. The sequence (∣qn∣)n≥0 is strictly increasing. Moreover, there exists a number ψ > 1such that ∣qn∣ ≥ ψn for all n ∈ N0.

iv. For every n ∈ N∣z − pn

qn∣ ≤ 1∣qn∣2 .

Proof. Part i is trivial, ii is Theorem 6.1. of [DaNo14], iii is Corollary 5.3. of [DaNo14],and iv is Theorem 1 of [La73].

Although our main results are stated for general irrational complex numbers, therewill be no loss of generality if we only work in F. Also, in view of Part ii of Proposition2.1, we will abuse our notation and denote the set F ∖Q(i) by F.

F. Schweiger defined in [Sc] a fibred system to be a pair (B,T ) where B is anon-empty set and T ∶ B → B is a function such that there exist an at most countablepartition of B, B(i)i∈I , such that for all i ∈ I the restriction of T to B(i) is injective.We borrow the terminology from the fibred system theory to better describe HCFs.

Define I = a ∈ Z[i] ∶ ∣a∣ ≥ √2. For every a ∈ Z[i] the cylinder of level 1 is

C1(a) = z ∈ F ∶ [z−1] = a .Note that C1(a) ≠ ∅ if and only if a ∈ I and that C1(a)a∈I gives a countable partitionof F (see Figure 1). Now, we can formally define the elements of the HCF as functionsfrom F onto I as follows

∀n ∈ N an(z) = a ⇐⇒ Tn−1(z) ∈ C1(a),


1

−0.5 0.5

−0.5

0.5

0

1 − i−1 − i

1 + i−1 + i

−2 − i

−1 − 2i −2i 1 − 2i

2 − i

2

2 + i

1 + 2i2 + 2i

3i

2i

−1 + 2i

−2 + i

−2 −3−3 − i

−3 + i

−2 + 2i −1 + 3i 1 + 3i

3 + i3

3 − i3 − 2i2 − 2i

−3i−2 − 2i −1 − 3i

Figure 1: Partition of F.

Figure 1: Partition C1(a) ∶ a ∈ I of F.

where T 0 ∶ F → F is the identity and Tn = T Tn−1 for n ∈ N. For a = (a1, . . . , an) ∈ Inor a ∈ IN the cylinder of level n, Cn(a), is the set

Cn(a) = z ∈ F ∶ a1(z) = a1, . . . , an(z) = an .A sequence a ∈ IN is admissible or valid if it is the HCF of some z ∈ F or, equivalently,if Cn(a) ≠ ∅ for every n ∈ N. We denote the set of admissible sequences by ΩHCF anddefine Φ ∶ ΩHCF → F by2

∀a ∈ ΩHCF Φ(a) = [0;a1, a2, a3, . . .].A maximal feasible set is a set F′ ⊆ F such that for some a ∈ ΩHCF and n ∈ N we have

Tn [Cn(a)] = F′.A maximal feasible set is regular if its interior is non-empty. A sequence a ∈ ΩHCF isregular if Tn[Cn(a)] is regular for every n and z ∈ F is regular if its HCF is regular. Forinstance, if w ∈ T1[C1(−2)], then (2 +w)−1 ∈ F, so

w ∈ (−2 + F−1) ∩ F = F−1 ∩ (2 − F) + 2 = F ∖D(1).On the other hand, if w ∈ F ∖ D(1), then −2 + w ∈ F−1 ∩ (2 − F), so [−2 + w] = −2 andw ∈ T [C1(−2)]. Similarly, we can show that T [C1(a)] = F if and only if ∣a∣ ≥ √

8. Figure

2Our function Φ deviates from from Schweiger’s. For him, Φ goes from the set B to the sequencespace.


2 illustrates F−1 and, with the help of the grid, the sets a+T [C1(a)] for ∣a∣ ≤ √8 can be

read.After dismissing the boundary, all the non-empty sets T [C1(a)] are of the form ijFk

for some j, k ∈ 1,2,3,4, where

F1 ∶= (F ∖ (D(−1) ∪D(−i))) , F2 ∶= (F ∖D(−1)) ,F3 ∶= (F ∖D(−1 − i)) , F4 = F.

Evidently, ijF4 = F4 for any j ∈ Z. By computing directly the inversion of some circles(v.gr. C(1+ i)−1 = C(1− i)), it can be shown inductively that these thirteen sets exhaustall the possibilities for the shapes that the interior of a regular maximal feasible set mayassume (cfr. [HiVa18]). 1

0−2 −1 1 2

−2−1

1

2

Figure 2: The set F−1 and the lines x = k, y = l for all k, l ∈ Z.

F1 F2 F3 F4

Figure 3: Sets F1, F2, F3, F4.

The previous observations and an iterative argument gives a simple necessary con-dition for a sequence to be admissible

E√8(1) ∶= a = (an)n≥1 ∈ Z[i]N ∶ ∀n ∈ N √8 ≥ ∣an∣ ⊆ ΩHCF.

Moreover, Tn[Cn(a)] = F holds for every n ∈ N whenever a ∈ E√8(1).


Hurwitz Continued Fractions share approximation properties with the regular con-tinued fractions. In particular, they provide quadratic approximations in terms of thedenominators (qn)n≥0.Proposition 2.2. There are absolute constants such that for any regular a ∈ ΩHCF withQ-pair ((pn)n≥0, (qn)n≥0) the following estimate holds

∀n ∈ N ∣Cn(a)∣ ≍ 1∣qn∣2 . (2)

Proof. Take a ∈ ΩHCF and n ∈ N. The inequality ∣Cn(a)∣ ≪ ∣qn∣−2 follows from Proposi-tion 2.1, iv. For ∣qn∣−2 ≪ ∣Cn(a)∣, take z,w ∈ Cn(a) and z′,w′ ∈ C such that

z = [0;a1, a2, . . . , an, an+1, an+2, . . .], w = [0;a1, a2, . . . , an, bn+1, bn+2 . . .],z′ = [an+1;an+2, an+3 . . .], w′ = [bn+1; bn+2, bn+3 . . .].

By regularity, we can choose z′,w′ such that z′ = 2w′ and ∣w′∣ ≤ 10, say, then

∣z −w∣ = 1∣qn∣2 ∣z′ −w′∣∣(z′ + qn−1

qn) (w′ + qn−1

qn)∣ ≥

1∣qn∣2 ∣w′∣(2∣w′∣ + 1)2 ≫ 1∣qn∣2 .In what follows, we will only be concerned with regular sequences without further

comment.For any a = (an)n≥1 ∈ ΩHCF define the family of Mobius transformations, (ta,n)n≥1,

by ∀n ∈ N ∀z ∈ C ta,n(z) = pn−1z + pnqn−1z + qn ,

where (pn)n≥0, (qn)n≥0 is the Q-pair of a. The maps ta,n are local inverses of Tn, asta,n Tn is the identity on Cn(a) and Tn ta,n is the identity on Tn[Cn(a)]. From asymbolic point of view, ta,n maps a word b to a1 . . . anb. Note that the restrictions ta,n ∶Tn[Cn(a)]→ Cn(a) are bi-Lipschitz, because they are bi-holomorphic. For reference, westate the following lemma.

Lemma 2.1. For any a ∈ ΩHCF (or any a = (a1, . . . , an) that can be extended to anelement of ΩHCF) and any n ∈ N the map ta,n ∶ Cn(a)→ F, which acts via

[0;a1, . . . , an, x1, x2, . . .]↦ [0;x1, x2, . . .],is bi-Lipschitz. Moreover, if b ∈ ΩHCF, m ∈ N, and Tn[Cn(a)] = Tm[Cm(b)], then themap from Cn(a) to Cm(b) given by

[0;a1, . . . , an, x1, x2, . . .]↦ [0; b1, . . . , bm, x1, x2, . . .]is bi-Lipschitz.

3 GENERALIZED JARNIK LEMMAS 8

For any L > 0 and any N ∈ N define the sets

EL(N) ∶= z = [0;a1, a2, . . .] ∈ F ∶ ∀n ∈ N0 ∣an+N ∣ ≥ I ,E′L ∶= z = [0;a1, a2, . . .] ∈ F ∶ lim inf

n→∞ ∣an∣ ≥ I .Lemma 2.2. For any L ∈ R, L ≥ 3, we have dimH EL(1) = dimH E

′L.

Proof. Let L ∈ R be at least 3. Since (EL(N))N≥1 is increasing and E′L = ⋃N≥1EL(N),

dimH E′L = lim

N→∞dimH EL(N).Thus, it suffices to show that for all N the relation dimH EL(N) ≤ dimH EL(1) holds.Let N ∈ N be at least 2. Define GL ∶= a ∈ Z[i] ∶ ∣a∣ ≥ L. Then,

EL(1) = Φ [GNL] ,

and we can write EL(N) as the countable union

EL(N) = ⋃a∈IN−1 Φ [(a ×GN

L) ∩ΩHCF ](some terms are empty). As a consequence, we obtain

dimH EL(N) = supa∈IN−1 dimH Φ [(a ×GN

L) ∩ΩHCF ] ,Take a ∈ IN−1 such that CN−1(a) ≠ ∅. In view of Lemma 2.1, the map

s ∶ Φ [(a ×GNL) ∩ΩHCF]→ TN−1[CN−1(a)] ∩Φ[GN

L]given by

s([0;a1, . . . , an, x1, x2, . . .]) = [0;x1, x2, x3, . . .].is bi-Lipschitz. The Hausdorff dimension invariance under bi-Lipschitz maps (Proposi-tion 3.3. of [Fa14]) gives

dimH Φ [(a ×GNL) ∩ΩHCF] ≤ dimH Φ[GN

L] = dimEL(1).Hence, by taking the supremum over a ∈ IN−1, dimH EL(N) ≤ dimH EL(1).3 Generalized Jarnık Lemmas

The main tools of the proof of Theorem 1.3 are two propositions, which we shall callGeneralized Jarnık Lemmas. They give bounds on the Hausdorff dimension of someCantor sets. In this section, our framework is a more restrictive version of stronglytree-like sets as defined in [KlWe10] (though we keep the name).


Definition 3.1. Let (X,d) be a complete metric space. A family of compact sets A isstrongly tree-like if A = ⋃∞n=0An where each Ak is finite, #A0 = 1, and

i. ∀A ∈ A ∣A∣ > 0,

ii. ∀n ∈ N ∀A,B ∈ An (A = B) ∨ (A ∩B = ∅),

iii. ∀n ∈ N ∀B ∈ An ∃A ∈ An−1 B ⊆ A,

iv. ∀n ∈ N ∀A ∈ An−1 ∃B ∈ An B ⊆ A,

v. dn(A) ∶= max∣A∣ ∶ A ∈ An→ 0 as n→∞.

The quantity dn(A) is the n-th stage diameter. The limit set of A, A∞, is

A∞ ∶= ∞⋂n=0 ⋃

A∈AnA.

As in [Ja28], if Y = Yjj∈J is an at most countable family of subsets of X and s ≥ 0,we write

Λs(Y) ∶=∑j∈J ∣Yj ∣s.

For X1,X2 ⊆X we denote the distance between them by

d(X1,X2) ∶= infd(y1, y2) ∶ y1 ∈ Y1, y2 ∈ Y2.Lemma 3.1 (First Generalized Jarnık Lemma). Let A be a tree-like family of compactsets (as defined above). Suppose that for some κ > 1

∀n ∈ N0 dn(A) ≤ 1

κn. (3)

Assume the existence of a sequence (Bn)n≥1 with 0 < Bn ≤ 1 for all n,

lim supn→∞

log log(B−1n )

logn< 1, (4)

and such that

∀n ∈ N ∀A ∈ An ∀Y,Z ∈ An+1 (Y ∪Z ⊆ A & Y ≠ Z Ô⇒ d(Y,Z) ≥ Bn∣A∣). (5)

If for s > 0 there exists some c > 0 such that

∀X ∈ 2A (#X < +∞ & A∞ ⊆ ⋃A∈XA Ô⇒ ΛsX > c) , (6)

then dimH A∞ ≥ s.


Proof. Keep the statement’s notation. Let G be a finite open cover of the compact setA∞. First, we associate to each element of G a compact AG ∈ A. For any G ∈ G letn ∈ N be maximal with respect to the property

∃A ∈ Am G ⊆ A,and let AG be the corresponding A ∈ An. By definition of AG, there are A′,A′′ ∈ An+1such that A′ ∩G ≠ ∅ and A′′ ∩G ≠ ∅, so

∣G∣ ≥ d(A′,A′′) ≥ Bn∣AG∣.Second, take A ∈ An such that A = AG for some G ∈ G and consider G ∈ G ∶ AG = A =G1, . . . ,Gr. Given ε > 0, we can take max∣G∣ ∶ G ∈ G so small that forces n to be solarge that κεnBn < 1 (by (4)), and then

∣A∣s ≤ 1

Bsn

r∑j=1 ∣Gj ∣s <

1

κεnBsn

r∑j=1 ∣Gj ∣s−ε <

r∑j=1 ∣Gj ∣s−ε.

In view of (5), we havec < ΛsX ≤ Λs−εG,

and taking infima, dimH A∞ > s − ε. Since ε > 0 was arbitrary, dimH A∞ ≥ s.Rather than a direct application of Lemma 3.1, we will change (6) for the stronger

condition ∀n ∈ N ∀A ∈ An ∑A′⊆A

A′∈An+1∣A′∣s ≥ ∣A∣s. (7)

To see that (7) implies (6) take a finite cover X ⊆ A of A∞. Without loss of generality,assume that the elements of X are disjoint by pairs. Let n be the maximal integersatisfying An ∩ X ≠ ∅ and B ∈ An ∩ X. By pairwise disjointness, if A ∈ An−1 satisfiesB ⊆ A, then all the sets B′ ∈ An with B′ ⊆ A also belong to X. Then, after replacing thesecompact sets B′ ∈ An with A, we obtain a new finite cover X′ such that ΛsX ≥ ΛsX

′.Repeating the argument we eventually arrive to X′ = A0, so (6) holds.

Lemma 3.2 (Second Generalized Jarnık Lemma). Let A be a family of compact setssatisfying conditions i., iii., iv., v. of the definition of strongly tree-like structure andsuch that the each Ak, k ≥ 1, is at most countable. Let s > 0.

If for every k ∈ N and every A ∈ Ak∑B

∣B∣s ≤ ∣A∣s,where the sum runs along the sets B satisfying B ∈ Ak+1 and B ⊆ A, then dimH A∞ ≤ s.Proof. For any k ≥ 1 we have Λs(Ak) ≤ Λs(A0) < +∞, because

Λs(Ak) = ∑A∈Ak

∣A∣s = ∑B∈Ak−1 ∑A∈Ak

A⊆B∣A∣s ≤ ∑

B∈Ak−1∣B∣s ≤ Λs(Ak−1).

Therefore, since every Ak covers A∞ and dk(A)→ 0 as k →∞, dimH A∞ ≤ s.

4 PRELIMINARY LEMMAS 11

4 Preliminary Lemmas

The Generalized Jarnık Lemmas allow us to bound some sets built through HCF re-strictions. With these sets, we will be able to approximate

E ∶= z = [0;a1, a2, a3, . . .] ∈ F ∶ limn→∞ ∣an∣ =∞

from the inside and from the outside. The outer approximation is obtained taking thenumbers in F whose elements have absolute value satisfying a uniform lower bound. Theinner approximation is done by considering complex numbers whose elements tend toinfinity at a certain rate.

For any L,M ∈ R>0 define

EML ∶= z = [0;a1, a2, a3, . . .] ∈ F ∶ ∀n ∈ N L ≤ ∣an∣ ≤M .Lemma 4.1. 1. For every L ≥ √

8

1 ≤ limM→∞ dimH E

ML .

2. For any positive number I let EL be defined as in Corollary 1.4, then

limI→∞dimH EL ≤ 1.

For any z ∈ C write ∥z∥ = max∣Rz∣, ∣Iz∣, so ∣z∣ ≍ ∥z∥.Lemma 4.2. Let f, g ∶ N → R>0 such that 3 ≤ f(n) ≤ g(n) for every positive integer n.For any n ∈ N, any a = (a1, . . . , an) ∈ Z[i]n such that

∀j ∈ [1..n] f(j) ≤ ∥aj∥ ≤ g(j),and any b ∈ Z[i] with f(n + 1) ≤ ∣b∣ ≤ g(n + 1) we have that

dimH (Cn(a) ∩Ef,g) = dimH (Cn+1(ab) ∩Ef,g) .Proof. Keep the theorem’s notation and let c ∈ Z[i] satisfy f(n + 1) ≤ ∥c∥ ≤ g(n + 1).Using Lemma 2.1 can see that (Cn+1(ab) ∩Ef,g) and (Cn+1(ac) ∩Ef,g) have the sameHausdorff dimension. We arrive at the desired conclusion by writing Cn(a) ∩ Ef,g asa countable union, like we did in the proof of Lemma 2.1. The details are left to thereader.

Lemma 4.3. Let f, g ∶ N → R>0 be functions such that for some fixed c with 0 < c < 1we have

√8 ≤ f(n) < (1 − c)g(n) and g(n)2 ≤ en for all n ∈ N. Define the sets

Ef,g ∶= [0;a1, a2, a3, . . .] ∈ F ∶ ∀n ∈ N f(n) ≤ ∥an∥ ≤ g(n) .Then, dimH Ef,g = 1.

5 PROOFS OF MAIN RESULTS 12

5 Proofs of Main Results

Proof of Theorem 1.3. From E ⊆ E′L for L ∈ N≥3 and the first part of Lemma 4.1, we

obtaindimH E ≤ 1.

Let f and g be as in the second part of Lemma 4.3, then Ef,g ⊆ E and the aforementionedlemma implies

1 ≤ dimH Ef,g ≤ dimH E.

Remark.. With the notation of Theorem 1.3, we also get that the set

Ef = z = [0;a1, a2, . . .] ∶ ∀n ∈ N f(n) ≤ ∣an∣has Hausdorff dimension 1, for Ef,g ⊆ Ef ⊆ E.

Proof of Corollary 1.4. The monotonicity of the limit is clear, because L↦ dimH EL isnon-increasing. Let ε > 0. By Lemma 4.3, we can take L = L(ε) ∈ N and M =M(L, ε) ∈ Nbe such that

1 − ε ≤ dimH EML ≤ dimH EL ≤ 1 + ε.

Since an irrational complex number belongs to BadC if and only if its HCF is bounded,EML ⊆ EL ∩BadC ⊆ EL and the result follows.

6 Proofs of Preliminary Lemmas

6.1 Proof of Lemma 4.1

Proof of Lemma 4.1. i. Take L ≥ √8. Note that for sufficiently large M > L we have

#b ∈ Z[i] ∶ L ≤ ∣b∣ ≤M ≥M2.

Let n be a positive integer and a ∈ Z[i]n be such that

∀j ∈ 1, . . . , n L ≤ ∣aj ∣ ≤M.

Set γ3 = γ1/γ2 with 0 < γ1 < γ2 the constants implied in (2). For any β > 0 and anylarge M we have

∑L≤∣b∣≤M ∣Cn+1(ab)∣β ≥ γβ1(M + 2)2β ∑

L≤∣b∣≤M1∣qn∣2β ≥ γβ1M

2

γβ2 (M + 2)2βγβ2∣qn∣2β

≥ M2−2βγβ3(1 + 2M

)2β ∣Cn(a)∣2β.The coefficient of ∣Cn(a)∣2β is at least 1 if and only if

(2 − 2β) logM + β log γ3 − 2β log (1 + 2

M) ≥ 0,

6 PROOFS OF PRELIMINARY LEMMAS 13

which, after rearranging the terms, is seen to be equivalent to

β ≤ 2 logM

2 logM + 2 log (1 + 2M

) − log γ3< 1.

Since the quotient tends to 1 as M →∞, we can take β = β(M) < 1 as close to 1 aswe please by considering a large M . Hence, in view of the First Generalized JarnıkLemma, we conclude that dimH E

ML ≥ 1 + O((logM)−1). The implied constant

depends on L.

ii. For any L ≥ √8 and any ε > 0

∑∣b∣≥L ∣Cn+1(a b)∣1+ε ≤ γ1+ε2∣qn∣2+2ε ∑∣b∣≥L1

∣b + qn−1qn

∣2+2ε ≪γ1+ε2∣qn∣2+2ε ∑L≤∣b∣

1∣b∣2+2ε , .with γ1, γ2 as before; hence,

∑∣b∣≥L ∣Cn+1(a b)∣1+ε ≪ 1∣qn∣2+2ε ∫∞

L

dr

r1+2ε = 1∣qn∣2+2ε 1

2εL2ε≪ε

1

L2ε∣Cn(a)∣1+ε.

Since L−2ε → 0 as L→∞, the Second Jarnık Generalized Lemma yields that for anylarge integer L

dimH EL ≤ 1 + ε.6.2 Proof of Lemma 4.3

We start with an observation which is stems from the triangle inequality.

Proposition 6.1. Let α,β ∈ F such that ∣a1(α)∣ ≥ √8 and ∣a1(β)∣ ≥ √

8. Then, thereexists an absolute constant c1 > 0 such that for any a, b ∈ Z[i], a ≠ b,

∣(a + α) − (b + β)∣ > c1.Proof of Lemma 4.3. Let a ∈ Z[i]n be such that

∀j ∈ 1, . . . , n f(j) ≤ ∥aj∥ ≤ g(j).Let b, c ∈ Z[i] satisfy f(n + 1) ≤ ∥b∥, ∥c∥ ≤ g(n + 1). Let ((pn)n≥0, (qn)n≥0) be theQ-pair of ab, and p′n+1, q′n+1 the last terms of the Q-pair of ac. Any β, γ ∈ F withf(n + 2) ≤ ∥a1(β)∥, ∥a1(γ)∥ ≤ g(n + 2) satisfy

d (Cn+1(a r) ∩E√8,Cn+1(a s) ∩E√8) ≥ ∣βpn + pn+1βqn + qn+1 − γpn + p

′n+1

γqn + q′n+1 ∣= 1∣qn∣2 ∣c + γ − (b + β)∣

∣(b + z + qn−1qn

) (c +w + qn−1qn

)∣≫ 1∣qn∣2 1(g(n) − 2)2 ≫ 1

g(n)2 ∣Cn+1(a) ∩E√8∣ .

REFERENCES 14

As in the proof of Lemma 4.1, we have

∑f(n+1)≤∥b∥≤g(n+1) ∣Cn+1(ab) ∩E√8∣β ≥ g(n)2−2β γ

β3 (1 − f(n)

g(n))(1 + 2

g(n))2β∣Cn(a) ∩E√8∣β

for any a ∈ Z[i]n with f(j) ≤ ∥aj∥ ≤ g(j), 1 ≤ j ≤ n. The coefficient of ∣Cn(a)∣β is at least1 if and only if

2 log g(n) + log (1 − f(n)g(n))

2 log g(n) − log γ3 + log (1 + 2g(n)) ≥ β.

Because of f ≤ (1 − c)g, the left-hand side tends to 1 as n → ∞ and, by the FirstGeneralized Jarnık Lemma, dimH Ef,g ≥ 1. The inequality dimH Ef,g ≤ 1 follows fromthe second part of Lemma 4.1 and Lemma 4.2.

References

[DaNo14] S. G. Dani and A. Nogueira, Continued fractions for complex numbers andvalues of binary quadratic forms. Trans. Amer. Math. Soc. 366 (2014), 3553 - 3583.

[DoKr03] M.M. Dodson and S. Kristensen Hausdorff dimension and Diophantine ap-proximation. Fractal geometry and applications: a jubilee of Benot Mandelbrot.Part 1, 305347, Proc. Sympos. Pure Math., 72, Part 1, Amer. Math. Soc., Provi-dence, RI, 2004.

[Fa14] K. Falconer Fractal Geometry. Mathematical foundations and applications. Thirdedition. John Wiley & Sons, Ltd., Chichester, 2014.

[Gon18] G. Gonzalez Robert, Purely Periodic and Transcendental Complex ContinuedFractions. (2018) arXiv:1805.08007 [math.NT]

[Goo41] I.J. Good, The fractional dimensional theory of continued fractions, Proc. Cam-bridge Philos. Soc. 37, (1941). 199228.

[HiVa18] G. Hiary and J. Vandehey Calculations of the invariant measure for HurwitzContinued Fractions. (2018) arXiv:1805.10151 [math.NT]

[Hu87] A. Hurwitz, Uber die Entwicklung complexer Grossen in Kettenbruche (Ger-man), Acta Math. 11 (1887), 187200.

[Ja28] V. Jarnık, Zur metrischen Theorie der diophantischen Approximationen, PraceMat.-Fiz. 36 (1928-29), pp. 91106.

[KlWe10] D. Kleinbock and B. Weiss, Modified Schmidt Games and Diophantine Ap-proximation with Weights, Adv. Math. 223 (2010), no. 4, 12761298.

REFERENCES 15

[KTV06] S. Kristensen, R. Thorn, and S. Velani, Diophantine Approximation and BadlyApproximable Sets. Adv. Math. 203 (2006), no. 1, 132169.

[La73] R.B. Lakein, Approximation properties of some complex continued fractions,Monatshefte fur Mathematik 77 (1973), 396 - 403.

[Sc] F. Schweiger Multidimensional Continued Fractions. Oxford: Oxford UniversityPress, 2000.

Gerardo Gonzalez RobertDepartment of MathematicsAarhus UniversityNy Munkegade 118, Aarhus C 8000, [email protected]

On a theorem of Bengoechea on equivalent numbers


Abstract

Let x, y be real irrational numbers and suppose their regular continued fractionexpansion is x = [a0;a1, a2, . . .], y = [b0; b1, b2, . . .]. A classical result of Serrettells us that there exists γ ∈ GL(2,Z) such that γx = y (acting through Mobiustransformations) if and only if for some s, t ∈ N we have an+t = bn+s for all naturalnumbers n. Recently, P. Bengoechea gave an estimate for t in terms of γ. Inthis paper, we show a theorem equivalent to Bencoechea’s result using a geometriccontinued fraction perspective.

1 Introduction

For any sequence of integers (an)n≥0 with an ≥ 1 for n ≥ 1 we adopt the usual notation

[a0;a1, a2, . . .] ∶= a0 + 1

a1 + 1

a2 + 1⋱,

and similarly when the sequence is finite. We refer to the previous expression as a regularcontinued fraction. Continued fractions are a classical topic in elementary number theory,for they possess important properties which allow to obtain non-trivial conclusions fromthe number they represent. For example, a real number is irrational if and only if itscontinued fraction is infinite. While infinite continued fractions expansions are unique,every non-zero rational number has exactly two expansions one ending in 1 and the otherone ending in an integer at least 2.

Consider the group GL(2,Z) acting on R ∪ ∞ via Mobius transformations; that is

∀γ = (a bc d

) ∈ GL(2,Z) ∀x ∈ R γx ∶= ax + bcx + d.

A famous theorem of J.A. Serret characterizes the orbits of GL(2,Z)/R in terms of theircontinued fraction expansion.

1

2 GEOMETRY OF CONTINUED FRACTIONS 2

Theorem 1.1 (J.A. Serret, 1866). V Let x, y be two real numbers. Then, x, y belongto the same orbit of GL(2,Z)/R if and only if x, y ∈ Q or if their continued fractionexpansion satisfy

x = [a0;a1, a2, . . . , as−1, as, as+1, . . .], y = [b0; b1, b2, . . . , bt−1, as, as+1, . . .].In a recent paper, P. Bengoechea gave an estimate for s in terms of γ. Namely, she

showed the following theorem.

Theorem 1.2 (P. Bengoechea, 2015). Let γ ∈ GL(2,Z) and let r be the length of thecontinued fraction of γ−1[∞]. Then, for each x ∈ R there is a natural number s ≤ r + 3such that

x = [a0;a1, a2, . . .], γx = [b0; b1, . . . , bt−1, as, as+1, as+2, . . .].In her proof, P. Bengoechea’s studied carefully a sequence of matrices in GL(2,Z) that

naturally arises from the continued fraction expansion of x. Relying on different tools,namely the geometry of continued fractions, we show a result equivalent to Theorem 1.2.

Theorem 1.3. Let GL(2,Z) act on R through Mobius transformations and let L be thelength of the continued fraction expansion of γ(0) ∈ Q. Then, for every x ∈ R ∖Q thereexists s ∈ N such that s ≤ L + 3 and

x = [a0;a1, a2, a3, . . .], γx = [b0, b1, . . . , bn, as, as+1, as+2, . . .].2 Geometry of Continued Fractions

In 1895, Felix Klein suggested a geometric approach to the study of continued fractions.For a detailed account of the geometry of continued fractions, we refer the reader tocitekarpenkov.

For any pair of integers m,n denote its greatest common divisor by gcd(m,n) ≥ 1.

Definition 2.1. The integer length of two points x = (x1, x2),y = (y1, y2) ∈ Z2,1d(x,y), is

1`(x,y) ∶= gcd(x1 − y1, x2 − y2).Let x,y,z ∈ Z2 be three different points. The angle θ =∠xzy is the ordered triple (x,z,y)and its integer sine is

1 sin(∠xzy) = ∣det(x − z,y − z)∣1`(x,z)1`(z,y) .

3 TWO ELEMENTARY LEMMAS 3

For any linearly independent x,y ∈ R2 with x ∈ Q2 write L+x ∶= tx ∶ t ≥ 0, L+y ∶= tx ∶t ≥ 0. The sail of L+x and L+y , S = S(x,y), is the boundary of the set

co(a, b) ∈ Z2 ∖ (0,0) ∶ ∃rx, ry ≥ 0 ∃t ∈ [0,1] (a, b) = rxx + t(ryy − rxx). (1)

In general, S is an infinite broken line and hence is homeomorphic to R. If we assume thatx ∈ Q2, then S and L+x eventually coincide. Moreover, since S and R are homeomorphic,we can call V0 the unique extreme point of the set in (1) lying on L+x, and successivelydenote by Vn the vertices of S.

Definition 2.2. Let x,y ∈ R2 be linearly independent with x ∈ Q. The LLS’ sequenceof x,y is the sequence of integers (an)n≥0 given by

∀n ∈ N a2n = 1`(Vn, Vn+1), a2n+1 = 1 sin(∠VnVn+1Vn+2).Remark. The LLS and LLS’ sequences coincide up to a shift and multiplication by -1 ofthe indices.

The following result is a restatement of Theorems 3.1. and 3.4. in [Ka13].

Theorem 2.1 (F. Klein, 1895). Let α > 1 be an irrational number. The sequence ofvertices of the angle θ =∠((1,0), (0,0), (1, α)), (Vn)n≥0 satisfies

∀n ∈ N0 Vn = (q2n−2, p2n−2).If α = [a0;a1, a2, . . .], the LLS’-sequence of θ is (an)n≥0. The sequence of vertices θ′ =∠((1,0), (0,0), (1, α)), (Wn)n≥0 satisfies

∀n ∈ N0 Wn = (q2n−1, p2n−1).If α = [a0;a1, a2, . . .], the LLS’-sequence of θ is (an)n≥0. A similar result holds for 0 <α < 1, in this case Vn = (q2n, p2n), Wn(q2n+1, p2n+1). If α = [0;a1, a2, . . .], then the LLS’sequence of θ is (an)n≥1.3 Two Elementary Lemmas

For the terminology regarding Farey sequences, see [NMZ91].

Lemma 3.1. Let r, s, u, v be integers such that rv − su = 1 and v, s ≥ 1. Then, rs and u

v

admit a regular continued fraction expansion of length n and m, respectively, such that∣n −m∣ ≤ 1.

4 PROOF OF THE MAIN THEOREM 4

Proof. Recall that that for any A,B,C,D ∈ Z satisfying B,D ≥ 1 and AD −BC = 1 thenumbers A

B and CD are adjacent in the the Farey sequence of order maxB,D (citenimozu,

p.301). Thus, the numbers adjacent to the regular continued fraction pkqk= [a0;a1, . . . , ak],

ak ≥ 2, in the Farey sequence of order qk are

pk−1qk−1 = [a0;a1, . . . , ak−1], (ak − 1)pk−1 + pk−2(ak − 1)qk−1 + qk−2 = [a0;a1, . . . , ak−1, ak − 1,2].

The lemma follows after applying the previous observation to the the rs or u

v dependingon which of the inequalities s > v or v > s hold.

Now we recall a well known continued fraction identity.

Lemma 3.2. Let α be a positive real number with continued fraction α = [a0;a1, a2, . . .] >0, then

−α = ⎧⎪⎪⎨⎪⎪⎩[−a0 − 1; 1, a1 − 1, a2, a3, . . .] if a1 ≠ 1,[−a0 − 1; 1, a2 + 1, a3, . . .] if a1 = 1.

4 Proof of the Main Theorem

Let us first note that for any n ∈ Z we have that

(1 n0 1

)(a bc d

) = (a + nc c + ndc d

) .Hence, since translations by integers affect only the first term in the continued fractionexpansion, we will not lose any generality by assuming that b < 0 < d and ac ≤ 0. Moreover,if the determinant is 1, then we must have ac = 0 and max∣a∣, ∣c∣ = 1. In any case, directverification with Lemma 3.2 give the result. Moreover, using the transformation x ↦ −xand Lemma 2, the result for matrices with determinant −1 follows from the one formatrices with positive determinant.

Proof. The preceding discussion tells us that we do not lose generality by assuming thatdetγ = 1 and

γ = (a bc d

) , b < 0 < d, c < 0 < a. (2)

Note that by the invariance of the LLS sequence under the action of GL(2,Z), we showthat the tails of the sails γ[Lx] and Sy eventually coincide.

Write Lα = t(1, α) ∶ t ∈ R for any α ∈ R. Let x be an irrational number and y = γx,so γ[Lx] = Ly where

γ = (d cb a

) .Consider L1 = L b

dand L2 ∶= La

c. We will also call Sx the sail of the angle (0,1), (0,0),(1, x).


Case I: 0 < x, y. Our assumptions imply that the angle formed by (d, b), (0,0) and(1, y) lies entirely in the semi-plane (x1, x2) ∈ R2 ∶ x1 ≥ 1. As a consequence, we musthave that (1,0) ∈ γ[Sx] for otherwise there should be a = (a1, a2),b = (b1, b2) ∈ N ×Z and0 < t < 1 such that ∥ta + (1 − t)b∥ < ∥(1,0)∥ = 1.

But a1, b1 ≥ 1 makes the previous inequality impossible. As a consequence, the tail of thebroken line γ[Sx] starting from (1,0) is Sy. The broken starting segment of γ[Sx] from(d, b) to (1,0) is the reflection on the x axis of S b

d. Therefore, since its LLS sequence

gives the continued fraction expansion of bd , we conclude that [aL+1;aL+2, aL+3, . . .] is a

tail of the expansion of y.

−4 −3 −2 −1 1 2 3 4 5 6

−4

−3

−2

−1

1

2

3

4

5

6

7

0

(d, b)

(c, a)

(1, y)

Figure 1: Case I. 0 < x, y.

Case II: y < 0 < x. Consider the integer points (Q,P ) and (Q′, P ′) with

Q = maxq ∈ Z≤0 ∶ ∃p ∈ N (q, p) ∈ γ[Sx] ,Q′ = minq ∈ N ∶ ∃p ∈ Z (q, p) ∈ γ[Sx] ,


and let P,P ′ be the corresponding integers. If Q = 0, then P = 1 (i.e. (0,1) ∈ γ[Sx])and the tail of Sx starting from (0,1) coincides with the S′−y. Assume that Q > 0 and let0 < α∗ < 1 be such that (0, α∗) is the intersection of γ[Sx] with the vertical axis. Let L′ bethe infinite broken line obtained from the intersection of γ[Sx] and the second quadrantand replacing the segment (P,Q), (0, α∗) with (P,Q), (0,1). We obtain the sail of theangle (−1, y), (0,0), (0,1), which gives the continued fraction of −y and, hence, of y. Wedo the same for (P ′,Q′), and we conclude that [xL+2;xL+3, xL+4, . . .] is a tail of y.

−4 −3 −2 −1 1 2 3 4 5 6

−4

−3

−2

−1

1

2

3

0

(d, b)

(c, a)

(1, y)

Figure 2: Case II. y < 0 < x.

Case III: x, y < 0. We start by noting that for any m ∈ Z we have that

γτm = (a am + bc cm + d) = (a bm

c dm) y = γτm(x −m).

Hence, take m ∈ N large such that c, d′ ∶= dm < 0 < a, b ∶= bm and note that x −m < x < 0.An argument similar to the one in the previous case gives the result.

In general, let γ ∈ GL(2,Z) have detγ = 1. By considering compositions τ−nγ for n ∈ Nwe can take γ into the previous treated cases. Moreover, since translations only affectthe first term of the continued fraction, the conclusions on the tails hold.

To conclude the result for this case, we must estimate the length of the continuedfraction of b′/d′. This follows, however, directly from Lemma 3.1 and ad−bc = a(d+mc)−(b +ma)c = 1.

5 FURTHER COMMENTS 7

−4 −3 −2 −1 1 2 3 4 5 6

−4

−3

−2

−1

1

2

3

4

0

(d, b)

(c, a)

(1, y)

Figure 3: Case III. y, x < 0.

5 Further Comments

References

[Be16] P. Bengoechea, On a theorem of Serret on continued fractions. Revista de la RealAcademia de Ciencias Exactas, Fısicas y Naturales. Serie A. Matematicas, 110 (2)(2016) 379-384.

[Ka13] O. Karpenkov, Geometry of continued fractions. Algorithms and Computation inMathematics, 26. Springer, Heidelberg, 2013.

[NMZ91] H. Montgomery, I. Niven, H. Zuckerman, An Introduction to the Theory ofNumbers. Fifth edition. John Wiley & Sons, Inc., New York, 1991.

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